Compare Potential Energy - Middle School Physical Science

1. A cart moves on a track through three labeled positions shown below: Position A at $h=0\,\text{m}$, Position B at $h=8\,\text{m}$, and Position C at $h=0\,\text{m}$. Ignoring friction, at which position(s) is the cart's gravitational potential energy the same?
   Answer: Positions A and C This question tests understanding that gravitational potential energy changes as an object's position changes vertically—specifically, that PE increases with height and decreases when height decreases, and is the same at equal heights. Position in Earth's gravitational field determines gravitational potential energy through the height variable in PE = mgh: when an object is at higher position (larger h), it has more gravitational PE because h is larger (more energy stored), and when at lower position (smaller h), it has less PE. For this cart motion on the track: At Positions A and C (both h = 0 m, PE = mg*0 = 0), PE is the same (minimum), while at Position B (h = 8 m, PE = mg*8, maximum), it's higher; ignoring friction, PE depends only on height, so same h means same PE regardless of location on track. Choice C is correct because it identifies Positions A and C as having the same PE, accurately noting they share the same height (h=0 m). Choice D is wrong because it includes B, but B has different h (8 m) so different PE; choice A says only B, but B is unique; choice B says only A, ignoring C at same h. The position-PE connection is fundamental to understanding gravitational potential energy: (1) PE is energy of position (where you are in gravitational field matters), (2) higher positions have more PE (climbing stores energy by increasing h), (3) lower positions have less PE (descending releases energy by decreasing h), and (4) PE can convert to other forms (primarily KE when falling: PE → KE maintains total energy). Applications include roller coasters where multiple points at the same height have equal PE, allowing energy conservation analysis, and water reservoirs at equal elevations having the same PE per unit mass.
2. A skateboarder rolls along a level sidewalk at a constant height of $h=2\text{ m}$ above a reference level. What happens to the skateboarder's gravitational potential energy as they move forward along the level sidewalk?
   Answer: It stays constant because height does not change. This question tests understanding that gravitational potential energy changes as an object's position changes vertically—specifically, that PE increases with height and decreases when height decreases. Position in Earth's gravitational field determines gravitational potential energy through the height variable in PE = mgh: when an object is at higher position (larger h), it has more gravitational PE because h is larger (more energy stored), and when at lower position (smaller h), it has less PE. As the skateboarder rolls along the level sidewalk, they maintain constant height h = 2 m above the reference level, so their gravitational potential energy remains PE = mg(2) throughout the horizontal motion—PE doesn't change because h doesn't change, even though horizontal position changes. The key insight is that gravitational PE depends only on vertical position (height h), not on horizontal position (x or y coordinates): moving north, south, east, or west at constant height doesn't change PE = mgh because h stays the same. Choice C is correct because it accurately states PE stays constant when height does not change, recognizing that horizontal motion at constant elevation doesn't affect gravitational PE. Choice A incorrectly claims PE increases just because the skateboarder is moving, confusing motion with height change or mixing PE with KE; Choice B wrongly connects friction (which affects KE by slowing motion) with PE, when PE depends only on height not speed; Choice D suggests horizontal position affects gravitational PE, contradicting the fact that only vertical position (height) matters for PE = mgh. The position-PE connection is fundamental to understanding gravitational potential energy: (1) PE is energy of position (where you are in gravitational field matters), (2) higher positions have more PE (climbing stores energy by increasing h), (3) lower positions have less PE (descending releases energy by decreasing h), and (4) PE can convert to other forms (primarily KE when falling: PE → KE maintains total energy). This principle explains why driving on a level highway doesn't change your car's gravitational PE (constant h), but driving up a mountain pass increases PE (increasing h) requiring more fuel, while coming down converts that PE to KE (must brake to control speed) or regenerative braking in electric vehicles can recover that PE as electrical energy.
3. An object of mass $m$ moves from Position A at $h=8\text{ m}$ to Position B at $h=3\text{ m}$ (same reference level). What is the object's change in gravitational potential energy, $\Delta PE = PE_B - PE_A$?
   Answer: $\Delta PE = mg(3-8)=-5mg$ This question tests understanding that gravitational potential energy changes as an object's position changes vertically—specifically, that PE increases with height and decreases when height decreases. Position in Earth's gravitational field determines gravitational potential energy through the height variable in PE = mgh: when an object is at higher position (larger h), it has more gravitational PE because h is larger (more energy stored), and when at lower position (smaller h), it has less PE. As the object moves from Position A (h = 8 m) to Position B (h = 3 m), it descends by Δh = 3 - 8 = -5 m (negative because moving downward), so its gravitational potential energy decreases from PEA = mg(8) = 8mg to PEB = mg(3) = 3mg, giving change ΔPE = PEB - PEA = 3mg - 8mg = -5mg J. The negative sign indicates PE decreased, which makes sense because the object moved to a lower position where it has less gravitational PE—this lost PE typically converts to kinetic energy if the object is falling freely. Choice B is correct because it properly calculates ΔPE = mg(3-8) = -5mg, using the correct formula ΔPE = mg(hfinal - hinitial) with negative result indicating PE decrease for downward motion. Choice A reverses the subtraction as mg(8-3) = +5mg, giving positive ΔPE when it should be negative for descent; Choice C incorrectly adds heights mg(8+3) = 11mg instead of finding the difference; Choice D claims ΔPE = 0 for downward motion, contradicting the fundamental principle that PE changes with height regardless of direction. The position-PE connection is fundamental to understanding gravitational potential energy: (1) PE is energy of position (where you are in gravitational field matters), (2) higher positions have more PE (climbing stores energy by increasing h), (3) lower positions have less PE (descending releases energy by decreasing h), and (4) PE can convert to other forms (primarily KE when falling: PE → KE maintains total energy). Understanding that ΔPE is negative for downward motion helps explain energy conversions: falling objects speed up (PE → KE), water flowing downhill in rivers has kinetic energy from lost PE, and avalanches gain destructive kinetic energy as snow loses PE descending the mountain.
4. An elevator lifts a person of mass $m$ from the ground floor (Position A, $h=0\text{ m}$) to the 10th floor (Position B, $h=30\text{ m}$). Which expression correctly gives the increase in the person's gravitational potential energy?
   Answer: $\Delta PE = mgh = mg\cdot 30\text{ m}$ This question tests understanding that gravitational potential energy changes as an object's position changes vertically—specifically, that PE increases with height and decreases when height decreases. Position in Earth's gravitational field determines gravitational potential energy through the height variable in PE = mgh: when an object is at higher position (larger h), it has more gravitational PE because h is larger (more energy stored), and when at lower position (smaller h), it has less PE. Moving upward increases PE (climbing stairs, lifting object, rising elevator all gain gravitational PE as they gain height), while moving downward decreases PE (falling, descending, lowering all lose gravitational PE as they lose height), with the change in PE calculated as ΔPE = mg(hfinal - hinitial) = mgΔh (positive if going up, negative if going down). For this upward motion: As the person rises from ground level (h = 0 m, reference where PE = 0) to h = 30 m, their gravitational potential energy increases from PE₀ = mg(0) = 0 J to PEfinal = mg(30) = 30mg J—the person gained PE = 30mg J during the lift, which represents the work done by the elevator against gravity (work = mg × 30 m). Choice B is correct because it properly calculates ΔPE = mgh = mg·30 m for the position change (increase). Choice A has wrong sign: positive to negative, as if descending; Choice C uses wrong formula (m/g instead of mg); Choice D uses wrong height (10 m instead of 30 m). The position-PE connection is fundamental to understanding gravitational potential energy: (1) PE is energy of position (where you are in gravitational field matters), (2) higher positions have more PE (climbing stores energy by increasing h), (3) lower positions have less PE (descending releases energy by decreasing h), and (4) PE can convert to other forms (primarily KE when falling: PE → KE maintains total energy). Applications: (1) hydroelectric dams store water at height (high PE), release it to fall (PE → KE → turbine rotation → electricity), (2) pumped storage: pump water uphill when electricity cheap (store as PE), release downhill when electricity needed (PE → KE → generate electricity), (3) roller coasters: lift cart to top (add PE), release (PE → KE creating speed for thrilling ride), (4) trebuchets/catapults: raise counterweight (store PE), release (PE → KE → projectile motion).
5. A person of mass $m$ climbs stairs from the first floor at height $h=0\,\text{m}$ to the second floor at $h=3\,\text{m}$ and then to the third floor at $h=6\,\text{m}$. How does the person's gravitational potential energy change from the first floor to the third floor (take the first floor as the reference level)?
   Answer: It increases by $mg(6\,\text{m})$ because height increases and $PE=mgh$. This question tests understanding that gravitational potential energy changes as an object's position changes vertically—specifically, that PE increases with height and decreases when height decreases. Position in Earth's gravitational field determines gravitational potential energy through the height variable in PE = mgh: when an object is at higher position (larger h), it has more gravitational PE because h is larger (more energy stored), and when at lower position (smaller h), it has less PE. As the person climbs from the first floor (h = 0 m, reference point where PE = 0) to the third floor (h = 6 m), their gravitational potential energy increases from PE₀ = mg(0) = 0 J to PEfinal = mg(6) = 6mg J—the person gained PE = 6mg J during the climb, which represents the work done against gravity to lift them (work = force × distance = mg × 6 m = same 6mg J). Choice B is correct because it accurately states PE increases by mg(6 m) when the person rises from h = 0 to h = 6 m, correctly applying ΔPE = mgΔh = mg(6-0) = mg(6 m). Choice A reverses the relationship: claims PE decreases when climbing, opposite of actual PE = mgh relationship where PE increases with height; Choice C claims only speed changes PE, when actually gravitational PE depends only on height (PE = mgh), not speed; Choice D incorrectly suggests only the last step matters, when actually the total height change from start to finish determines ΔPE. The position-PE connection is fundamental to understanding gravitational potential energy: (1) PE is energy of position (where you are in gravitational field matters), (2) higher positions have more PE (climbing stores energy by increasing h), (3) lower positions have less PE (descending releases energy by decreasing h), and (4) PE can convert to other forms (primarily KE when falling: PE → KE maintains total energy). Applications include hydroelectric dams storing water at height (high PE) to release for electricity generation, and even everyday activities like climbing stairs where you store energy as PE that could be released if you jump down.
6. A ball is thrown upward from $h=0\,\text{m}$ to $h=5\,\text{m}$ and then returns to $h=0\,\text{m}$. Ignoring air resistance, how does its gravitational potential energy change during the motion?
   Answer: It increases on the way up, is maximum at $h=5\,\text{m}$, then decreases back to its starting value at $h=0\,\text{m}$. This question tests understanding that gravitational potential energy changes as an object's position changes vertically—specifically, that PE increases with height and decreases when height decreases. Position in Earth's gravitational field determines gravitational potential energy through the height variable in PE = mgh: when an object is at higher position (larger h), it has more gravitational PE because h is larger (more energy stored), and when at lower position (smaller h), it has less PE. As the ball moves: starting at h = 0 m, PE = 0; rising to h = 5 m, PE increases to mg(5) = 5mg (maximum at peak); falling back to h = 0 m, PE decreases back to 0—the PE increases during upward motion, reaches maximum at the highest point, then decreases during downward motion, returning to its starting value when the ball returns to starting height. Choice A is correct because it accurately describes PE increasing on the way up (as height increases), being maximum at h = 5 m (the highest point), then decreasing back to starting value at h = 0 m, properly tracking how PE follows height throughout the motion. Choice B confuses PE with KE, claiming PE decreases going up because speed decreases—actually PE increases going up while KE decreases; Choice C incorrectly claims PE stays constant, ignoring that PE varies with height during the motion; Choice D absurdly claims PE increases in both directions, violating energy conservation. The position-PE connection is fundamental to understanding gravitational potential energy: (1) PE is energy of position (where you are in gravitational field matters), (2) higher positions have more PE (climbing stores energy by increasing h), (3) lower positions have less PE (descending releases energy by decreasing h), and (4) PE can convert to other forms (primarily KE when falling: PE → KE maintains total energy). A thrown ball demonstrates complete energy transformation: initial KE from throw converts to PE while rising (KE → PE), reaches all PE at peak (KE = 0, PE = max), then PE converts back to KE while falling (PE → KE), with total energy conserved throughout.
7. Two identical carts (same mass) are on different tracks. Cart A is at height $h=8\,\text{m}$ and Cart B is at height $h=3\,\text{m}$, measured from the same reference level. Which comparison is correct for their gravitational potential energies?
   Answer: Cart A has more gravitational potential energy because it is at a greater height. This question tests understanding that gravitational potential energy changes as an object's position changes vertically—specifically, that PE increases with height and decreases when height decreases. Position in Earth's gravitational field determines gravitational potential energy through the height variable in PE = mgh: when an object is at higher position (larger h), it has more gravitational PE because h is larger (more energy stored), and when at lower position (smaller h), it has less PE. Moving upward increases PE (climbing stairs, lifting object, rising elevator all gain gravitational PE as they gain height), while moving downward decreases PE (falling, descending, lowering all lose gravitational PE as they lose height), with the change in PE calculated as ΔPE = mg(hfinal - hinitial) = mgΔh (positive if going up, negative if going down). For comparing the two carts: Cart A at h = 8 m has PE_A = mgh_A = mg(8) = 8mg J, while Cart B at h = 3 m has PE_B = mgh_B = mg(3) = 3mg J. Since both carts have identical mass m and experience same gravity g, the cart at greater height has more PE: PE_A = 8mg > PE_B = 3mg, with Cart A having 8mg - 3mg = 5mg J more potential energy than Cart B. The height difference alone determines the PE difference when masses are equal. Choice C is correct because it accurately states Cart A has more gravitational PE due to its greater height—at h = 8 m versus h = 3 m, Cart A has higher position in gravitational field and thus more stored energy. Choice A reverses the relationship, claiming Cart B (lower position) has more PE when actually lower height means less PE; Choice B incorrectly states equal PE just because masses are equal, ignoring that different heights give different PE even for same mass; Choice D suggests PE depends on speed rather than height, confusing kinetic energy (speed-dependent) with potential energy (position-dependent). The position-PE connection is fundamental to understanding gravitational potential energy: (1) PE is energy of position (where you are in gravitational field matters), (2) higher positions have more PE (climbing stores energy by increasing h), (3) lower positions have less PE (descending releases energy by decreasing h), and (4) PE can convert to other forms (primarily KE when falling: PE → KE maintains total energy). This comparison illustrates why height matters for energy storage: two identical objects at different heights have different energy potentials—the higher one could do more work if released (more PE to convert), which explains why water towers are built tall (more PE for water pressure) and why higher diving boards allow more spectacular dives (more PE → KE → athletic maneuvers).
8. A rock of mass $m$ falls from a cliff at height $h=12\text{ m}$ (Position A) to a ledge at height $h=5\text{ m}$ (Position B). What is the change in gravitational potential energy, $\Delta PE = PE_B - PE_A$?
   Answer: $\Delta PE = mg(5-12)=-7mg$ This question tests understanding that gravitational potential energy changes as an object's position changes vertically—specifically, that PE increases with height and decreases when height decreases. Position in Earth's gravitational field determines gravitational potential energy through the height variable in PE = mgh: when an object is at higher position (larger h), it has more gravitational PE because h is larger (more energy stored), and when at lower position (smaller h), it has less PE. Moving downward decreases PE (falling, descending, lowering all lose gravitational PE as they lose height), with the change in PE calculated as ΔPE = mg(hfinal - hinitial) = mgΔh (negative if going down, as Δh negative). For this downward motion: As the rock falls from h=12 m (PE_A = mg(12)) to h=5 m (PE_B = mg(5)), ΔPE = mg(5 - 12) = -7mg J—this loss in PE converts to kinetic energy (rock speeds up), demonstrating energy conservation. Choice B is correct because it properly calculates ΔPE = mg(5-12)=-7mg for the position change. Choice A has wrong sign (positive +7mg, as if ascending); Choice C adds heights incorrectly (17mg); Choice D claims no change, ignoring height decrease. The position-PE connection is fundamental to understanding gravitational potential energy: (1) PE is energy of position (where you are in gravitational field matters), (2) higher positions have more PE (climbing stores energy by increasing h), (3) lower positions have less PE (descending releases energy by decreasing h), and (4) PE can convert to other forms (primarily KE when falling: PE → KE maintains total energy). Applications: (1) hydroelectric dams store water at height (high PE), release it to fall (PE → KE → turbine rotation → electricity), (2) pumped storage: pump water uphill when electricity cheap (store as PE), release downhill when electricity needed (PE → KE → generate electricity), (3) roller coasters: lift cart to top (add PE), release (PE → KE creating speed for thrilling ride), (4) trebuchets/catapults: raise counterweight (store PE), release (PE → KE → projectile motion).
9. A ball is dropped from rest from a balcony at height $h=8\,\text{m}$ above the ground (ground is the reference level, $h=0$). As the ball falls toward the ground, what happens to its gravitational potential energy?
   Answer: It decreases because the height decreases. This question tests understanding that gravitational potential energy changes as an object's position changes vertically—specifically, that PE increases with height and decreases when height decreases. Position in Earth's gravitational field determines gravitational potential energy through the height variable in PE = mgh: when an object is at higher position (larger h), it has more gravitational PE because h is larger (more energy stored), and when at lower position (smaller h), it has less PE. Moving upward increases PE (climbing stairs, lifting object, rising elevator all gain gravitational PE as they gain height), while moving downward decreases PE (falling, descending, lowering all lose gravitational PE as they lose height), with the change in PE calculated as ΔPE = mg(hfinal - hinitial) = mgΔh (positive if going up, negative if going down). For the falling ball: Initially at balcony (h = 8 m), PE_initial = mgh = mg(8); as it falls toward ground (h decreasing from 8 m toward 0 m), PE continuously decreases because height h is decreasing in PE = mgh; at ground (h = 0 m), PE_final = mg(0) = 0 J—the ball's gravitational PE decreases throughout the fall from maximum at release to zero at ground. Choice C is correct because it states PE decreases as height decreases, properly recognizing that falling objects lose gravitational PE as they descend. Choice A incorrectly claims PE increases because ball speeds up (wrong—confuses PE with KE; speeding up means KE increases while PE decreases during fall), Choice B claims PE stays constant due to constant gravity (wrong—g is constant but h changes so PE = mgh changes), and Choice D claims PE is greatest at ground where it stops (wrong—PE is minimum/zero at ground level h = 0, not maximum). The decreasing PE during the fall converts to increasing kinetic energy: total mechanical energy (PE + KE) remains constant, so as PE decreases from mg(8) to 0, KE increases from 0 to mg(8), demonstrating energy transformation during free fall.
10. A skateboarder rolls down a ramp from a platform at $h=4\,\text{m}$ to the ground at $h=0\,\text{m}$. Which statement correctly describes the skateboarder's gravitational potential energy during the descent?
    Answer: It decreases because height decreases; potential energy is converted to other forms like kinetic energy. This question tests understanding that gravitational potential energy changes as an object's position changes vertically—specifically, that PE increases with height and decreases when height decreases. Position in Earth's gravitational field determines gravitational potential energy through the height variable in PE = mgh: when an object is at higher position (larger h), it has more gravitational PE because h is larger (more energy stored), and when at lower position (smaller h), it has less PE. Moving upward increases PE (climbing stairs, lifting object, rising elevator all gain gravitational PE as they gain height), while moving downward decreases PE (falling, descending, lowering all lose gravitational PE as they lose height), with the change in PE calculated as ΔPE = mg(hfinal - hinitial) = mgΔh (positive if going up, negative if going down). For the skateboarder descending: rolling from platform (h = 4 m) down to ground (h = 0 m), the gravitational potential energy decreases from PE₁ = mgh₁ = mg(4) = 4mg J to PE₂ = mg(0) = 0 J. The skateboarder loses PE = 4mg J during descent, with this lost PE converting to kinetic energy (speed increases) as they roll down—energy transforms from PE to KE while total energy remains constant. At the bottom, all the initial PE has become KE, making the skateboarder move fastest at the lowest point. Choice B is correct because it accurately states PE decreases as height decreases during descent and correctly identifies that this PE converts to other forms like kinetic energy—the complete explanation of energy transformation during the roll down. Choice A claims PE increases because speed increases, confusing cause and effect: speed increases because PE decreases and converts to KE, not the reverse; Choice C states PE stays same because gravity is constant, missing that PE = mgh changes when h changes even though g is constant; Choice D suggests PE depends only on mass and cannot change, ignoring the height dependence in PE = mgh. The position-PE connection is fundamental to understanding gravitational potential energy: (1) PE is energy of position (where you are in gravitational field matters), (2) higher positions have more PE (climbing stores energy by increasing h), (3) lower positions have less PE (descending releases energy by decreasing h), and (4) PE can convert to other forms (primarily KE when falling: PE → KE maintains total energy). Skateboard ramps perfectly demonstrate energy conversion: starting high with PE and no speed, rolling down converts PE → KE (gaining speed), reaching bottom with maximum KE and minimum PE—understanding this helps skaters judge speed: higher starting point means more PE to convert, thus higher speed at bottom.
11. A person climbs stairs from the first floor at height $h=0\,\text{m}$ to the second floor at $h=3\,\text{m}$ and then to the third floor at $h=6\,\text{m}$. Which statement best describes how the person's gravitational potential energy (PE) changes as they go up?
    Answer: PE increases because height increases; it is greatest at $h=6\,\text{m}$. This question tests understanding that gravitational potential energy changes as an object's position changes vertically—specifically, that PE increases with height and decreases when height decreases. Position in Earth's gravitational field determines gravitational potential energy through the height variable in PE = mgh: when an object is at higher position (larger h), it has more gravitational PE because h is larger (more energy stored), and when at lower position (smaller h), it has less PE. As the person climbs from h=0 m to h=3 m and then to h=6 m, their gravitational potential energy increases stepwise: starting at PE=0 (reference), to PE=mg(3) at the second floor, and finally to PE=mg(6) at the third floor, with the greatest PE at the highest point (h=6 m) because that's the maximum height achieved. Choice B is correct because it accurately states PE increases when the object rises (h increases) and is greatest at h=6 m, properly connecting position in the gravitational field to PE value. Choice A is wrong because it reverses the relationship, claiming PE decreases when moving upward, opposite of the actual PE = mgh relationship; Choice C suggests PE doesn't change with position, contradicting the fundamental dependence PE = mgh where h is position; Choice D identifies maximum PE at the lowest position, when actually PE is maximum where h is maximum (highest point). The position-PE connection is fundamental to understanding gravitational potential energy: (1) PE is energy of position (where you are in gravitational field matters), (2) higher positions have more PE (climbing stores energy by increasing h), (3) lower positions have less PE (descending releases energy by decreasing h), and (4) PE can convert to other forms (primarily KE when falling: PE → KE maintains total energy). Applications include elevators storing PE by lifting people to higher floors, which represents work done against gravity that could be recovered if descending.
12. Two hikers have the same mass. Hiker 1 stands on a hill at $h=12\,\text{m}$ above the reference level. Hiker 2 stands on a platform at $h=12\,\text{m}$ above the same reference level. Which statement is correct about their gravitational potential energies?
    Answer: They have the same PE because they are at the same height. This question tests understanding that gravitational potential energy changes as an object's position changes vertically—specifically, that PE increases with height and decreases when height decreases. Position in Earth's gravitational field determines gravitational potential energy through the height variable in PE = mgh: when an object is at higher position (larger h), it has more gravitational PE because h is larger (more energy stored), and when at lower position (smaller h), it has less PE. For the hikers: Both at h=12 m with same mass m, so PE=mg(12) for each, identical regardless of whether on hill or platform, as PE depends only on height above reference. Choice C is correct because it accurately states they have the same PE since they are at the same height, connecting position (vertical) to PE value. Choice A and Choice B wrongly suggest steepness or man-made nature affects PE, but only h matters; Choice D claims dependence on path, but PE is path-independent, depending only on final position. The position-PE connection is fundamental to understanding gravitational potential energy: (1) PE is energy of position (where you are in gravitational field matters), (2) higher positions have more PE (climbing stores energy by increasing h), (3) lower positions have less PE (descending releases energy by decreasing h), and (4) PE can convert to other forms (primarily KE when falling: PE → KE maintains total energy). Applications include comparing energies at same altitudes, like airplanes or mountains, where PE is the same at equal heights.
13. A person of mass $m$ climbs stairs from the first floor (Position A, $h=0\,\text{m}$) to the second floor (Position B, $h=3\,\text{m}$) and then to the third floor (Position C, $h=6\,\text{m}$). If gravitational potential energy is $PE=mgh$, which position has the greatest gravitational potential energy?
    Answer: Position C (third floor, $h=6\,\text{m}$) This question tests understanding that gravitational potential energy changes as an object's position changes vertically—specifically, that PE increases with height and decreases when height decreases. Position in Earth's gravitational field determines gravitational potential energy through the height variable in PE = mgh: when an object is at higher position (larger h), it has more gravitational PE because h is larger (more energy stored), and when at lower position (smaller h), it has less PE. For this upward motion of climbing stairs: As the person rises from Position A (h = 0 m, PE = mg*0 = 0) to Position B (h = 3 m, PE = mg*3) and then to Position C (h = 6 m, PE = mg*6), the gravitational potential energy increases progressively with each floor, reaching the maximum at the highest position (C) because PE is directly proportional to h—the person gains PE = mgΔh during the climb, which represents the work done against gravity. Choice C is correct because it accurately identifies the greatest PE at the highest position (h=6 m), where h is maximum, so PE = mgh is largest. Choice D is wrong because it claims all positions have the same PE due to constant mass, contradicting the fundamental dependence of PE on height (h) in PE = mgh, where different h means different PE even if m is the same; choices A and B incorrectly select lower heights as having greater PE, reversing the PE-height relationship. The position-PE connection is fundamental to understanding gravitational potential energy: (1) PE is energy of position (where you are in gravitational field matters), (2) higher positions have more PE (climbing stores energy by increasing h), (3) lower positions have less PE (descending releases energy by decreasing h), and (4) PE can convert to other forms (primarily KE when falling: PE → KE maintains total energy). Applications include roller coasters that lift carts to high positions to store PE for the ride and hydroelectric dams that store water at height for energy generation.
14. An elevator lifts a person of mass $m$ from the ground floor ($h=0\text{ m}$) to the 10th floor ($h=30\text{ m}$). Ignoring friction, which statement best describes the work done by the motor on the person?
    Answer: The motor does work equal to $\Delta PE = mg(30-0)=30mg$. This question tests understanding that gravitational potential energy changes as an object's position changes vertically—specifically, that PE increases with height and decreases when height decreases. Position in Earth's gravitational field determines gravitational potential energy through the height variable in PE = mgh: when an object is at higher position (larger h), it has more gravitational PE because h is larger (more energy stored), and when at lower position (smaller h), it has less PE. As the elevator lifts the person from ground floor (h = 0 m, reference point where PE = 0) to the 10th floor (h = 30 m), the person's gravitational potential energy increases from PE₀ = mg(0) = 0 J to PEfinal = mg(30) = 30mg J—the person gained ΔPE = PEfinal - PE₀ = 30mg - 0 = +30mg J during the lift. By the work-energy theorem, the motor must do work equal to this PE increase: W = ΔPE = 30mg J, which represents the energy transferred from the motor's electrical energy to the person's gravitational PE (ignoring friction means no energy lost to heat). Choice A is correct because it accurately states the motor does work equal to ΔPE = mg(30-0) = 30mg, properly connecting the work done to lift an object with its gain in gravitational PE. Choice B incorrectly claims no work is done, ignoring that lifting against gravity requires work even without horizontal motion; Choice C has the wrong calculation with negative result mg(0-30) = -30mg, when work done by motor lifting is positive; Choice D wrongly suggests work depends on speed not height change, but work against gravity equals mgh regardless of how fast you lift. The position-PE connection is fundamental to understanding gravitational potential energy: (1) PE is energy of position (where you are in gravitational field matters), (2) higher positions have more PE (climbing stores energy by increasing h), (3) lower positions have less PE (descending releases energy by decreasing h), and (4) PE can convert to other forms (primarily KE when falling: PE → KE maintains total energy). Elevators demonstrate this principle daily: motors do work to lift people (increasing their PE), and in modern buildings with regenerative systems, descending elevators can convert PE back to electrical energy as people's PE decreases, recovering some energy rather than wasting it as heat in brakes.
15. A cart rolls down a track from $h=15\,\text{m}$ to $h=5\,\text{m}$. Which statement best explains what happens to the cart's gravitational potential energy during this descent?
    Answer: It decreases because height decreases, since $PE=mgh$. This question tests understanding that gravitational potential energy changes as an object's position changes vertically—specifically, that PE increases with height and decreases when height decreases. Position in Earth's gravitational field determines gravitational potential energy through the height variable in PE = mgh: when an object is at higher position (larger h), it has more gravitational PE because h is larger (more energy stored), and when at lower position (smaller h), it has less PE. For the cart: Rolling from h=15 m (higher PE) to h=5 m (lower PE), PE decreases as height decreases, with ΔPE=mg(5-15)=-10mg, converting to KE as it speeds up. Choice B is correct because it accurately states PE decreases when height decreases (h decreases), properly citing PE=mgh. Choice A confuses PE with KE, claiming increase due to speeding up; Choice C suggests constant PE despite height change; Choice D identifies greatest PE at lowest point, opposite of reality. The position-PE connection is fundamental to understanding gravitational potential energy: (1) PE is energy of position (where you are in gravitational field matters), (2) higher positions have more PE (climbing stores energy by increasing h), (3) lower positions have less PE (descending releases energy by decreasing h), and (4) PE can convert to other forms (primarily KE when falling: PE → KE maintains total energy). Applications include ramps or slides, where descending converts PE to KE for motion.
16. A pendulum bob swings from the left endpoint (Position A), down to the lowest point (Position B), and up to the right endpoint (Position C). The endpoints are $0.5\text{ m}$ above the lowest point, so $h_A=0.5\text{ m}$, $h_B=0\text{ m}$, and $h_C=0.5\text{ m}$ (with $h=0$ at the lowest point). Which statement about gravitational potential energy (PE) is correct?
    Answer: PE is the same at Positions A and C because they are at the same height. This question tests understanding that gravitational potential energy changes as an object's position changes vertically—specifically, that PE increases with height and decreases when height decreases. Position in Earth's gravitational field determines gravitational potential energy through the height variable in PE = mgh: when an object is at higher position (larger h), it has more gravitational PE because h is larger (more energy stored), and when at lower position (smaller h), it has less PE. For pendulum: The pendulum bob swings through positions: at left endpoint (h = 0.5 m above bottom), PE is high and KE is zero (momentarily at rest); swinging down toward bottom (h decreasing toward 0), PE decreases and converts to KE (bob speeds up); at bottom (h = 0, reference), PE = 0 (minimum) and KE maximum (fastest); continuing to right endpoint (h increasing back to 0.5 m), PE increases again and KE decreases (slows to stop at endpoint). The PE changes track the height changes perfectly: highest at endpoints (h = 0.5 m, maximum height, same for A and C), lowest at bottom (h = 0, minimum height), with smooth variation between these extremes as bob swings. Choice B is correct because it accurately states PE is the same at A and C since they are at the same height (h=0.5 m). Choice A confuses PE and KE: claims PE greatest at B because moving fastest, when speeding up means KE maximum while PE minimum; Choice C reverses: claims lowest at A and C, but actually highest there; Choice D claims PE constant, contradicting dependence on h. The position-PE connection is fundamental to understanding gravitational potential energy: (1) PE is energy of position (where you are in gravitational field matters), (2) higher positions have more PE (climbing stores energy by increasing h), (3) lower positions have less PE (descending releases energy by decreasing h), and (4) PE can convert to other forms (primarily KE when falling: PE → KE maintains total energy). Applications: (1) hydroelectric dams store water at height (high PE), release it to fall (PE → KE → turbine rotation → electricity), (2) pumped storage: pump water uphill when electricity cheap (store as PE), release downhill when electricity needed (PE → KE → generate electricity), (3) roller coasters: lift cart to top (add PE), release (PE → KE creating speed for thrilling ride), (4) trebuchets/catapults: raise counterweight (store PE), release (PE → KE → projectile motion).
17. A rock of mass $3\,\text{kg}$ falls from a cliff at $h=12\,\text{m}$ to a ledge at $h=7\,\text{m}$. Using $g\approx 10\,\text{m/s}^2$, what is the change in gravitational potential energy ($\Delta PE = PE_{\text{final}} - PE_{\text{initial}}$) for this fall?
    Answer: $\Delta PE = -150\,\text{J}$ This question tests understanding that gravitational potential energy changes as an object's position changes vertically—specifically, that PE increases with height and decreases when height decreases. Position in Earth's gravitational field determines gravitational potential energy through the height variable in PE = mgh: when an object is at higher position (larger h), it has more gravitational PE because h is larger (more energy stored), and when at lower position (smaller h), it has less PE. Moving upward increases PE (climbing stairs, lifting object, rising elevator all gain gravitational PE as they gain height), while moving downward decreases PE (falling, descending, lowering all lose gravitational PE as they lose height), with the change in PE calculated as ΔPE = mg(hfinal - hinitial) = mgΔh (positive if going up, negative if going down). For the falling rock: starting at cliff top (h = 12 m) with PE_initial = mgh₁ = (3 kg)(10 m/s²)(12 m) = 360 J, falling to ledge (h = 7 m) with PE_final = mgh₂ = (3 kg)(10 m/s²)(7 m) = 210 J. The change in PE is ΔPE = PE_final - PE_initial = 210 J - 360 J = -150 J, or using the formula: ΔPE = mgΔh = (3)(10)(7-12) = (3)(10)(-5) = -150 J. The negative sign indicates PE decreased during the fall, with 150 J of PE converting to kinetic energy as the rock gains speed. Choice B is correct because it accurately calculates ΔPE = -150 J for the rock falling from h = 12 m to h = 7 m, with negative sign properly indicating PE decrease during downward motion. Choice A shows +150 J with wrong sign, suggesting PE increases when falling, opposite of actual behavior; Choice C calculates -360 J, which is negative of initial PE but not the change (would require falling to h = 0); Choice D claims no change (0 J), ignoring that height decreased from 12 m to 7 m which must decrease PE. The position-PE connection is fundamental to understanding gravitational potential energy: (1) PE is energy of position (where you are in gravitational field matters), (2) higher positions have more PE (climbing stores energy by increasing h), (3) lower positions have less PE (descending releases energy by decreasing h), and (4) PE can convert to other forms (primarily KE when falling: PE → KE maintains total energy). Falling objects demonstrate energy conversion: initial PE at height converts to KE during fall, with the 150 J lost from PE becoming 150 J gained in KE (rock speeds up)—understanding this helps predict impact: higher falls mean more PE to convert, thus higher impact speeds and forces.
18. A box is lifted straight up from a shelf at $h=1\,\text{m}$ to a higher shelf at $h=4\,\text{m}$. Which expression gives the box's change in gravitational potential energy?
    Answer: $\Delta PE = mg(4-1)$ This question tests understanding that gravitational potential energy changes as an object's position changes vertically—specifically, that PE increases with height and decreases when height decreases. Position in Earth's gravitational field determines gravitational potential energy through the height variable in PE = mgh: when an object is at higher position (larger h), it has more gravitational PE because h is larger (more energy stored), and when at lower position (smaller h), it has less PE. As the box is lifted from the lower shelf (h = 1 m) to the higher shelf (h = 4 m), its gravitational potential energy increases: PEinitial = mg(1), PEfinal = mg(4), so ΔPE = PEfinal - PEinitial = mg(4) - mg(1) = mg(4-1) = 3mg—the positive change reflects energy added by lifting against gravity. Choice B is correct because it gives ΔPE = mg(4-1), properly expressing the change in PE as final minus initial heights multiplied by mg, which equals mg(3) = 3mg for the upward motion. Choice A reverses the subtraction mg(1-4), giving negative result for upward motion; Choice C adds heights mg(4+1) instead of finding the difference; Choice D incorrectly divides m by g instead of multiplying. The position-PE connection is fundamental to understanding gravitational potential energy: (1) PE is energy of position (where you are in gravitational field matters), (2) higher positions have more PE (climbing stores energy by increasing h), (3) lower positions have less PE (descending releases energy by decreasing h), and (4) PE can convert to other forms (primarily KE when falling: PE → KE maintains total energy). Warehouse forklifts demonstrate this daily: lifting boxes to high shelves requires work (adding PE = mgΔh), which represents stored energy that could be dangerous if boxes fall—safety protocols exist because that stored PE would convert to destructive KE in a fall.
19. A rock of mass $3\,\text{kg}$ falls from a cliff at height $h=12\,\text{m}$ to a ledge at height $h=4\,\text{m}$ (same reference level for both heights). Using $g\approx 10\,\text{m/s}^2$, what is the change in gravitational potential energy, $\Delta PE = PE_{\text{final}}-PE_{\text{initial}}$?
    Answer: $-240\,\text{J}$ This question tests understanding that gravitational potential energy changes as an object's position changes vertically—specifically, that PE increases with height and decreases when height decreases. Position in Earth's gravitational field determines gravitational potential energy through the height variable in PE = mgh: when an object is at higher position (larger h), it has more gravitational PE because h is larger (more energy stored), and when at lower position (smaller h), it has less PE. As the rock falls from the cliff (h = 12 m) to the ledge (h = 4 m), its gravitational potential energy decreases: PEinitial = mg(12) = (3 kg)(10 m/s²)(12 m) = 360 J; PEfinal = mg(4) = (3 kg)(10 m/s²)(4 m) = 120 J; so ΔPE = PEfinal - PEinitial = 120 J - 360 J = -240 J—the negative sign indicates PE decreased as the rock fell to lower position. Choice B is correct because it gives ΔPE = -240 J, accurately calculating the negative change in PE when falling from h = 12 m to h = 4 m, with the negative sign properly indicating a decrease in PE during downward motion. Choice A has wrong sign (+240 J), suggesting PE increases when falling; Choices C and D use 360 J instead of the actual change of 240 J, with C having wrong sign (+) and D having correct sign (-) but wrong magnitude. The position-PE connection is fundamental to understanding gravitational potential energy: (1) PE is energy of position (where you are in gravitational field matters), (2) higher positions have more PE (climbing stores energy by increasing h), (3) lower positions have less PE (descending releases energy by decreasing h), and (4) PE can convert to other forms (primarily KE when falling: PE → KE maintains total energy). When the rock falls, the 240 J of lost PE converts to kinetic energy, making the rock speed up as it falls—this PE → KE conversion powers everything from waterfalls to meteorite impacts.
20. A roller coaster cart moves through these heights above the lowest point: bottom $h=0\,\text{m}$, top of first hill $h=20\,\text{m}$, halfway down $h=10\,\text{m}$, then back to bottom $h=0\,\text{m}$. At which position is the cart's gravitational potential energy the greatest?
    Answer: At the top of the first hill ($h=20\,\text{m}$), because height is greatest. This question tests understanding that gravitational potential energy changes as an object's position changes vertically—specifically, that PE increases with height and decreases when height decreases. Position in Earth's gravitational field determines gravitational potential energy through the height variable in PE = mgh: when an object is at higher position (larger h), it has more gravitational PE because h is larger (more energy stored), and when at lower position (smaller h), it has less PE. Moving upward increases PE (climbing stairs, lifting object, rising elevator all gain gravitational PE as they gain height), while moving downward decreases PE (falling, descending, lowering all lose gravitational PE as they lose height), with the change in PE calculated as ΔPE = mg(hfinal - hinitial) = mgΔh (positive if going up, negative if going down). For roller coaster motion: At the bottom of the track (h = 0, reference), PE = 0 but KE is maximum (moving fast); as cart climbs to the top (h = 20 m), PE increases to mgh = m(10)(20) = 200m J while KE decreases (cart slows climbing); at the top, PE is maximum (highest point) while KE is minimum (slowest speed); then descending to h = 10 m halfway down, PE decreases to 100m J (half the peak) and KE increases (cart speeds up). The position changes throughout the ride create PE changes: highest PE at highest position (h = 20 m), lowest PE at lowest position (h = 0 m), with PE continuously varying between these extremes following the height. Choice C is correct because it correctly identifies maximum PE at highest position—the top of first hill at h = 20 m, where PE = mgh = m(10)(20) = 200m J, the greatest value among all positions. Choice A identifies maximum PE at lowest position (h = 0), when actually PE is minimum there (PE = 0) and confuses high speed (KE) with high PE; Choice B suggests halfway down has maximum PE, but h = 10 m gives PE = 100m J, which is less than at h = 20 m; Choice D claims PE doesn't change with position, contradicting the fundamental dependence PE = mgh where h is position. The position-PE connection is fundamental to understanding gravitational potential energy: (1) PE is energy of position (where you are in gravitational field matters), (2) higher positions have more PE (climbing stores energy by increasing h), (3) lower positions have less PE (descending releases energy by decreasing h), and (4) PE can convert to other forms (primarily KE when falling: PE → KE maintains total energy). Roller coasters demonstrate this perfectly: lift cart to top (add PE), release (PE → KE creating speed for thrilling ride), with the initial height determining maximum speed possible at bottom through energy conservation—understanding that position determines PE helps explain why the first hill is typically the tallest (maximum PE storage) and why subsequent hills must be lower (can't gain more energy than initially stored).

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