Net Force and Motion - Middle School Physical Science

A student graphs acceleration $a$ (y-axis) versus net force $F_{net}$ (x-axis) for the same cart. The points lie on a straight line that goes through the origin. What does this graph show?
Answer: Net force and acceleration are directly proportional for this cart (constant mass). This question tests understanding that a straight-line graph through the origin indicates a direct proportional relationship between variables. When graphing acceleration (y-axis) versus net force (x-axis) for constant mass, Newton's Second Law F_net = ma predicts a = (1/m)F_net—this is the equation of a straight line through the origin with slope 1/m. The straight line through the origin means: (1) when F_net = 0, acceleration = 0 (balanced forces produce no acceleration), and (2) acceleration increases linearly with net force—doubling force doubles acceleration, tripling force triples acceleration. The slope of this line equals 1/m (the reciprocal of the cart's mass): a steeper slope means smaller mass (easier to accelerate), while a gentler slope means larger mass (harder to accelerate). For example, if the slope is 0.5 s²/kg, then the cart's mass is 2 kg. Choice A is correct because it accurately identifies that the straight line through the origin shows net force and acceleration are directly proportional for constant mass. Choice B claims an inverse relationship (a ∝ 1/F_net), which would produce a curved hyperbola graph, not a straight line; Choice C suggests a threshold or minimum force requirement, which would show a line not passing through the origin but intercepting the x-axis at some positive value; Choice D claims constant acceleration regardless of force, which would produce a horizontal line, not a line with positive slope. This graphical analysis is a powerful tool in physics: plotting experimental data reveals the mathematical relationship between variables—a straight line through the origin is the clearest indicator of direct proportionality, confirming that Newton's Second Law accurately describes how forces cause acceleration.
A student gives the same ball different pushes on a smooth floor. During each push, the contact time is about the same, $\Delta t=0.5\,\text{s}$ . The student measures the ball's change in speed. Data: - $F_{\text{net}}=4\,\text{N}$ → $\Delta v=1\,\text{m/s}$ - $F_{\text{net}}=8\,\text{N}$ → $\Delta v=2\,\text{m/s}$ - $F_{\text{net}}=12\,\text{N}$ → $\Delta v=3\,\text{m/s}$ Which statement best describes the pattern between $F_{\text{net}}$ and the motion change?
Answer: As $F_{\text{net}}$ increases, $\Delta v$ increases in direct proportion (double the force gives double the speed change). This question tests understanding that net force and acceleration are directly proportional, as described by Newton's Second Law: F_net = m a, and since Δv = a Δt with fixed Δt, Δv is also proportional to F_net. The data show a direct proportional relationship between net force and change in speed: when net force doubles from 4 N to 8 N, Δv doubles from 1 to 2 m/s, and to 12 N (tripled from 4 N), Δv triples to 3 m/s—this linear pattern confirms Δv ∝ F_net for constant m and Δt. Looking at the data: the ratio F_net / Δv is constant at 4/1=4, 8/2=4, 12/3=4 (this relates to m / Δt, since Δv = (F_net / m) Δt), verifying the proportionality. Choice A is correct because it accurately describes the proportional relationship: as F_net increases, Δv increases in direct proportion, with doubling force giving double the speed change. Choice B is wrong because it claims Δv decreases as F_net increases, but the data show the opposite: Δv increases with F_net, not inversely. Collecting and analyzing force-motion data: (1) set up investigation with constant mass object, (2) apply different net forces (use force sensor or known applied force minus friction), (3) measure resulting acceleration for each force (motion sensor, or calculate from distance and time), (4) record data in table: F_net | a values, (5) graph data: F_net on x-axis, a on y-axis, (6) analyze pattern: should see straight line through origin showing proportionality, (7) calculate slope: slope = Δa/ΔF_net = 1/m, allows determining object's mass and extending to Δv predictions.
A student records the following data for the same cart: $F_{net}$ (N) | $a$ (m/s $^2$ ) 5 | 2.5 10 | 5.0 15 | 7.5 Which statement is best supported by the data?
Answer: Tripling the net force triples the acceleration (for the same cart). This question tests understanding that net force and acceleration are directly proportional, as described by Newton's Second Law: F_net = ma. The data show a direct proportional relationship between net force and acceleration: when net force increases, acceleration increases by the same factor (double the force → double the acceleration), and when net force decreases, acceleration decreases proportionally—this relationship is linear, meaning if you graph F_net (x-axis) vs acceleration (y-axis), you get a straight line through the origin, and the slope of that line equals 1/m (the reciprocal of the object's mass). This proportionality is Newton's Second Law: F_net = ma, which can be rearranged to a = F_net/m, showing that for constant mass, acceleration is directly proportional to net force. Looking at the data: when net force is 5 N, acceleration is 2.5 m/s², when net force increases to 10 N (doubled), acceleration increases to 5.0 m/s² (also doubled), and when net force increases further to 15 N (tripled from original), acceleration is 7.5 m/s² (also tripled)—this consistent doubling and tripling demonstrates perfect proportionality. The ratio F_net/a is constant throughout: 5/2.5 = 2, 10/5 = 2, 15/7.5 = 2 (this constant ratio equals the object's mass m = 2 kg in this example), confirming Newton's Second Law F_net = ma holds for all data points. Choice A is correct because it accurately identifies the proportional relationship: F_net ∝ a or F_net = ma / correctly interprets the data showing that doubling force doubles acceleration / properly recognizes the linear pattern in graph or constant ratio in table / correctly predicts that increasing net force will proportionally increase acceleration. Choice B is wrong because it claims an inverse relationship (larger force → smaller acceleration) when the data clearly show the opposite: as force increases from 5 N to 10 N to 15 N, acceleration increases from 2.5 to 5.0 to 7.5 m/s² (both increase together) / suggests there's no relationship or that acceleration is independent of force, when the data clearly show acceleration changes systematically with net force. Collecting and analyzing force-motion data: (1) set up investigation with constant mass object, (2) apply different net forces (use force sensor or known applied force minus friction), (3) measure resulting acceleration for each force (motion sensor, or calculate from distance and time), (4) record data in table: F_net | a values, (5) graph data: F_net on x-axis, a on y-axis, (6) analyze pattern: should see straight line through origin showing proportionality, (7) calculate slope: slope = Δa/ΔF_net = 1/m, allows determining object's mass from the data. Real investigations might show slight deviations from perfect line (measurement uncertainty, friction variations), but overall pattern should be clear: larger net forces produce larger accelerations proportionally—this relationship F_net = ma is one of the most fundamental in physics, describing how forces cause motion changes in everything from tiny molecules to planets, and collecting data to verify it experimentally (as students can do with carts, ramps, and force sensors) demonstrates how physics laws are not just theoretical but are actually observed patterns in nature that we can measure and test.
A student pushes the same 2 kg cart on a smooth track with different net forces and measures the cart's acceleration. The results are shown below. Force $F_{net}$ (N) | Acceleration $a$ (m/s $^2$ ) 0 | 0 5 | 2.5 10 | 5.0 15 | 7.5 What relationship between net force and acceleration is shown by the data?
Answer: Acceleration is directly proportional to net force; doubling $F_{net}$ doubles $a$ . This question tests understanding that net force and acceleration are directly proportional, as described by Newton's Second Law: F_net = ma. The data show a direct proportional relationship between net force and acceleration: when net force increases, acceleration increases by the same factor (double the force → double the acceleration), and when net force decreases, acceleration decreases proportionally—this relationship is linear, meaning if you graph F_net (x-axis) vs acceleration (y-axis), you get a straight line through the origin, and the slope of that line equals 1/m (the reciprocal of the object's mass). This proportionality is Newton's Second Law: F_net = ma, which can be rearranged to a = F_net/m, showing that for constant mass, acceleration is directly proportional to net force. Looking at the data: when net force is 5 N, acceleration is 2.5 m/s², when net force increases to 10 N (doubled), acceleration increases to 5.0 m/s² (also doubled), and when net force increases further to 15 N (tripled from original), acceleration is 7.5 m/s² (also tripled)—this consistent doubling and tripling demonstrates perfect proportionality. The ratio F_net/a is constant throughout: 5/2.5 = 2, 10/5 = 2, 15/7.5 = 2 (this constant ratio equals the object's mass m = 2 kg in this example), confirming Newton's Second Law F_net = ma holds for all data points. The straight line through the origin shows the proportional relationship clearly—the graph passes through (0,0) because zero net force produces zero acceleration (balanced forces), and increases linearly showing each additional Newton of force produces a consistent amount of additional acceleration. Choice B is correct because it accurately identifies the proportional relationship: F_net ∝ a or F_net = ma / correctly interprets the data showing that doubling force doubles acceleration / properly recognizes the linear pattern in graph or constant ratio in table / correctly predicts that increasing net force will proportionally increase acceleration. Choice A is wrong because it claims an inverse relationship (larger force → smaller acceleration) when the data clearly show the opposite: as force increases from 5 N to 10 N to 15 N, acceleration increases from 2.5 to 5.0 to 7.5 m/s² (both increase together). Collecting and analyzing force-motion data: (1) set up investigation with constant mass object, (2) apply different net forces (use force sensor or known applied force minus friction), (3) measure resulting acceleration for each force (motion sensor, or calculate from distance and time), (4) record data in table: F_net | a values, (5) graph data: F_net on x-axis, a on y-axis, (6) analyze pattern: should see straight line through origin showing proportionality, (7) calculate slope: slope = Δa/ΔF_net = 1/m, allows determining object's mass from the data. Real investigations might show slight deviations from perfect line (measurement uncertainty, friction variations), but overall pattern should be clear: larger net forces produce larger accelerations proportionally—this relationship F_net = ma is one of the most fundamental in physics, describing how forces cause motion changes in everything from tiny molecules to planets, and collecting data to verify it experimentally (as students can do with carts, ramps, and force sensors) demonstrates how physics laws are not just theoretical but are actually observed patterns in nature that we can measure and test.
A cart starts from rest and is pushed with different net forces for the same time interval, $\Delta t = 2\,\text{s}$ . The cart's mass stays the same in every trial. Measured accelerations: - $F_{\text{net}}=5\,\text{N}$ → $a=2.5\,\text{m/s}^2$ - $F_{\text{net}}=10\,\text{N}$ → $a=5.0\,\text{m/s}^2$ Using $\Delta v = a\Delta t$ , what is the cart's change in speed when $F_{\text{net}}=10\,\text{N}$ is applied for $2\,\text{s}$ ?
Answer: $10.0\,\text{m/s}$ This question tests understanding that net force and acceleration are directly proportional, as described by Newton's Second Law: F_net = m a, and how this connects to change in velocity via Δv = a Δt for constant time. The data show a direct proportional relationship between net force and acceleration: for constant mass, a = F_net / m, so Δv = (F_net / m) Δt, meaning Δv is also proportional to F_net when Δt is fixed. Looking at the data: for 5 N, a=2.5 m/s², Δv=2.5*2=5 m/s; for 10 N, a=5.0 m/s² (doubled), so Δv=5.0*2=10 m/s (also doubled)—this demonstrates the proportionality extends to velocity change. Choice C is correct because it correctly calculates Δv = a Δt = 5.0 * 2 = 10 m/s for F_net=10 N, matching the pattern where doubling force doubles acceleration and thus doubles Δv. Choice B is wrong because it suggests 5.0 m/s, perhaps confusing Δv with a itself or using the wrong time, but the formula clearly gives 10 m/s. Collecting and analyzing force-motion data: (1) set up investigation with constant mass object, (2) apply different net forces (use force sensor or known applied force minus friction), (3) measure resulting acceleration for each force (motion sensor, or calculate from distance and time), (4) record data in table: F_net | a values, (5) graph data: F_net on x-axis, a on y-axis, (6) analyze pattern: should see straight line through origin showing proportionality, (7) calculate slope: slope = Δa/ΔF_net = 1/m, allows determining object's mass from the data and predicting motion changes like Δv.
In a tug-of-war, two teams pull on a rope attached to a sled on ice. The net force on the sled is the difference between the pulls. A student measures the sled's acceleration for different net forces. $F_{net}$ (N) | $a$ (m/s $^2$ ) 2 | 0.5 4 | 1.0 6 | 1.5 If the net force is increased from 2 N to 6 N, how does the acceleration change?
Answer: It triples (from 0.5 m/s $^2$ to 1.5 m/s $^2$ ). This question tests understanding that net force and acceleration are directly proportional, as described by Newton's Second Law: F_net = ma. The data show a direct proportional relationship between net force and acceleration: when net force increases, acceleration increases by the same factor (double the force → double the acceleration), and when net force decreases, acceleration decreases proportionally—this relationship is linear, meaning if you graph F_net (x-axis) vs acceleration (y-axis), you get a straight line through the origin, and the slope of that line equals 1/m (the reciprocal of the object's mass). This proportionality is Newton's Second Law: F_net = ma, which can be rearranged to a = F_net/m, showing that for constant mass, acceleration is directly proportional to net force. Looking at the data: when net force is 2 N, acceleration is 0.5 m/s², when net force increases to 4 N (doubled), acceleration increases to 1.0 m/s² (also doubled), and when net force increases further to 6 N (tripled from original), acceleration is 1.5 m/s² (also tripled)—this consistent doubling and tripling demonstrates perfect proportionality. The ratio F_net/a is constant throughout: 2/0.5 = 4, 4/1 = 4, 6/1.5 = 4 (this constant ratio equals the object's mass m = 4 kg in this example), confirming Newton's Second Law F_net = ma holds for all data points. Choice B is correct because it correctly interprets the data showing that tripling force triples acceleration / properly recognizes the linear pattern in graph or constant ratio in table / correctly predicts that increasing net force will proportionally increase acceleration. Choice C is wrong because it claims an inverse relationship (larger force → smaller acceleration) when the data clearly show the opposite: as force increases from 2 N to 4 N to 6 N, acceleration increases from 0.5 to 1.0 to 1.5 m/s² (both increase together) / makes incorrect prediction: claims doubling force would triple or halve acceleration, when proportionality means doubling force doubles acceleration. Collecting and analyzing force-motion data: (1) set up investigation with constant mass object, (2) apply different net forces (use force sensor or known applied force minus friction), (3) measure resulting acceleration for each force (motion sensor, or calculate from distance and time), (4) record data in table: F_net | a values, (5) graph data: F_net on x-axis, a on y-axis, (6) analyze pattern: should see straight line through origin showing proportionality, (7) calculate slope: slope = Δa/ΔF_net = 1/m, allows determining object's mass from the data. Real investigations might show slight deviations from perfect line (measurement uncertainty, friction variations), but overall pattern should be clear: larger net forces produce larger accelerations proportionally—this relationship F_net = ma is one of the most fundamental in physics, describing how forces cause motion changes in everything from tiny molecules to planets, and collecting data to verify it experimentally (as students can do with carts, ramps, and force sensors) demonstrates how physics laws are not just theoretical but are actually observed patterns in nature that we can measure and test.
A $2\,\text{kg}$ cart on a track is pushed with a constant net force for $3\,\text{s}$ . The cart starts from rest. Trial A: $F_{\text{net}} = 5\,\text{N}$ Trial B: $F_{\text{net}} = 15\,\text{N}$ Which statement best compares the change in speed ( $\Delta v$ ) in Trial B to Trial A?
Answer: Trial B has three times the change in speed of Trial A. This question tests understanding that net force and acceleration are directly proportional (F_net = ma) and how this affects velocity change over time (Δv = aΔt). For Trial A: a = F_net/m = 5/2 = 2.5 m/s², so Δv = 2.5 × 3 = 7.5 m/s; for Trial B: a = F_net/m = 15/2 = 7.5 m/s², so Δv = 7.5 × 3 = 22.5 m/s—the velocity change in Trial B (22.5 m/s) is exactly three times that of Trial A (7.5 m/s). Since force in Trial B (15 N) is three times the force in Trial A (5 N), and acceleration is proportional to force, the acceleration is also three times greater (7.5 vs 2.5 m/s²), leading to three times the velocity change over the same time period. Choice C is correct because Trial B has three times the change in speed of Trial A—this follows directly from tripling the force, which triples the acceleration, which triples the velocity change for the same time duration. Choice A incorrectly claims one-third when it's actually three times; Choice B wrongly states equal changes when 22.5 ≠ 7.5 m/s; Choice D suggests nine times (perhaps squaring the force ratio) which doesn't match the linear relationship. This demonstrates how force affects motion over time: (1) larger force → larger acceleration (F = ma), (2) larger acceleration → faster velocity change (a = Δv/Δt), (3) for same time, velocity change is proportional to force, (4) tripling force triples final speed when starting from rest—this principle explains why more powerful engines produce faster acceleration.
A student pushes the same cart starting from rest for $t=2\,\text{s}$ each time. The net force is measured with a force sensor, and the cart's acceleration is measured with a motion sensor. Trial data: - $F_{\text{net}}=5\,\text{N}$ → $a=2.5\,\text{m/s}^2$ - $F_{\text{net}}=15\,\text{N}$ → $a=7.5\,\text{m/s}^2$ Assuming constant acceleration in each trial, how much farther does the cart travel in 2 s with $15\,\text{N}$ than with $5\,\text{N}$ ? (Use $d=\tfrac{1}{2}at^2$ .)
Answer: $10\,\text{m}$ This question tests understanding that net force and acceleration are directly proportional, as described by Newton's Second Law: F_net = m a, and how this affects distance traveled via d = (1/2) a t² for constant time. The data show a direct proportional relationship between net force and acceleration: since a = F_net / m, distance d = (1/2) (F_net / m) t², so d is proportional to F_net for fixed t and m. Looking at the data: for 5 N, a=2.5 m/s², d=(1/2)*2.5*4=5 m; for 15 N, a=7.5 m/s², d=(1/2)*7.5*4=15 m—the difference is 15-5=10 m, showing tripling force triples acceleration and thus triples distance. Choice B is correct because it correctly calculates the difference in distance as 10 m, based on the proportional relationship and the formula d=(1/2) a t². Choice A is wrong because it gives 5 m, perhaps calculating only the distance for 5 N or misapplying the formula without finding the difference. Collecting and analyzing force-motion data: (1) set up investigation with constant mass object, (2) apply different net forces (use force sensor or known applied force minus friction), (3) measure resulting acceleration for each force (motion sensor, or calculate from distance and time), (4) record data in table: F_net | a values, (5) graph data: F_net on x-axis, a on y-axis, (6) analyze pattern: should see straight line through origin showing proportionality, (7) calculate slope: slope = Δa/ΔF_net = 1/m, allows determining object's mass and predicting outcomes like distance differences.
A student wants to design an investigation to show how net force affects acceleration for a cart. Which plan would best test the relationship between $F_{net}$ and $a$ ?
Answer: Use the same cart, apply several different net forces, measure the acceleration each time, and compare how $a$ changes with $F_{net}$ . This question tests understanding of experimental design to investigate the relationship between net force and acceleration. To properly test how F_net affects acceleration, we need to: (1) control variables by keeping mass constant (same cart), (2) systematically vary the independent variable (net force), (3) measure the dependent variable (acceleration), and (4) analyze the pattern in the data. The key principle is isolating variables: to see how F_net affects a, we must keep mass constant while changing only the net force—if we change both force and mass simultaneously, we can't determine which factor caused the observed changes in acceleration. A good investigation would use the same cart (constant mass), apply several different measured net forces (using spring scales, known weights, or force sensors), measure the resulting acceleration for each force (using motion sensors, photogates, or video analysis), record data in a table, and graph a vs F_net to reveal the proportional relationship. Choice A is correct because it properly controls mass (same cart), systematically varies net force (several different forces), measures the outcome (acceleration), and compares results to find the relationship—this is proper experimental design. Choice B fails because it changes multiple variables (both mass and force), making it impossible to isolate the effect of force on acceleration; Choice C doesn't measure acceleration (the dependent variable we're investigating) and measures irrelevant properties instead; Choice D changes mass between trials (should be constant) and doesn't properly control or measure the net force. Good experimental design in physics requires: clear identification of variables, proper control of constants, systematic variation of the independent variable, accurate measurement of the dependent variable, and analysis of patterns in the data—this approach allows us to discover and verify fundamental relationships like Newton's Second Law.
A student applies different net forces to the same cart and records the acceleration: $F_{\text{net}}$ (N): 5, 10, 15 a ( $\text{m/s}^2$ ): 2, 4, 6 What happens to the acceleration when the net force increases from $5\,\text{N}$ to $15\,\text{N}$ ?
Answer: It triples from $2\,\text{m/s}^2$ to $6\,\text{m/s}^2$ . This question tests understanding that net force and acceleration are directly proportional, as described by Newton's Second Law: F_net = ma. The data show a direct proportional relationship: when net force is 5 N, acceleration is 2 m/s²; when force doubles to 10 N, acceleration doubles to 4 m/s²; when force triples to 15 N, acceleration triples to 6 m/s²—this consistent pattern confirms F_net ∝ a. Looking specifically at the change from 5 N to 15 N: the force increases by a factor of 3 (15/5 = 3), so the acceleration must also increase by a factor of 3, going from 2 m/s² to 6 m/s² (6/2 = 3). Choice B is correct because it accurately states that acceleration triples from 2 m/s² to 6 m/s² when force triples from 5 N to 15 N, maintaining the proportional relationship. Choice A incorrectly claims acceleration decreases when the data clearly show it increases; Choice C wrongly states acceleration stays constant at 2 m/s² when it actually changes to 6 m/s²; Choice D says acceleration only doubles to 4 m/s², but that's the value at 10 N, not 15 N. This proportional relationship means that for constant mass: (1) doubling force doubles acceleration, (2) tripling force triples acceleration, (3) the ratio F_net/a equals the object's mass (here 5/2 = 2.5 kg), (4) graphing the data produces a straight line through the origin—understanding this relationship allows us to predict motion changes for any applied force.
A student tests the same cart with several net forces and records acceleration. Which calculation would stay constant for every trial if the cart's mass does not change?
Answer: $F_{\text{net}}/a$ This question tests understanding that for constant mass, the ratio F_net/a remains constant according to Newton's Second Law: F_net = ma. Rearranging gives F_net/a = m, which means for the same cart (constant mass), every trial should yield the same value when dividing net force by acceleration—this ratio equals the cart's mass. For any data point, F_net/a gives the mass: if F_net = 10 N and a = 5 m/s², then m = 10/5 = 2 kg; if F_net = 20 N and a = 10 m/s², then m = 20/10 = 2 kg—the ratio stays constant at 2 kg regardless of the specific force or acceleration values. Choice C is correct because F_net/a equals mass, which remains constant for the same cart throughout all trials—this is the mathematical expression of Newton's Second Law rearranged to show what stays constant. Choice A (a/F_net) gives 1/m which varies inversely with mass but isn't the mass itself; Choice B (F_net + a) has no physical meaning and changes with each trial; Choice D (F_net × a) also varies with each trial and doesn't represent any fundamental quantity. Understanding constant ratios in physics: (1) F_net/a = m (Newton's Second Law), (2) this ratio reveals the object's inertial mass, (3) larger mass means larger ratio (more force needed per unit acceleration), (4) this constancy allows us to identify objects by their force-acceleration behavior—just as density (mass/volume) identifies materials, the F/a ratio identifies an object's resistance to acceleration.
A student applies the same net force of 10 N to two different carts on the same smooth track. Cart X has mass 2 kg and Cart Y has mass 5 kg. Which statement best compares their accelerations?
Answer: Cart X accelerates more because with the same $F_{net}$ , the smaller mass has the larger acceleration. This question tests understanding that for the same net force, acceleration is inversely proportional to mass, as described by Newton's Second Law: a = F_net/m. When the same net force (10 N) is applied to different masses, the smaller mass experiences greater acceleration because acceleration equals force divided by mass. For Cart X: a = F_net/m = 10 N / 2 kg = 5 m/s²; for Cart Y: a = F_net/m = 10 N / 5 kg = 2 m/s²—Cart X accelerates 2.5 times faster than Cart Y because it has 2.5 times less mass. This inverse relationship between mass and acceleration (for constant force) means that doubling the mass halves the acceleration, tripling the mass reduces acceleration to one-third, and so on. The same 10 N force that gives the 2 kg cart an acceleration of 5 m/s² only gives the 5 kg cart an acceleration of 2 m/s². Choice B is correct because it accurately identifies that Cart X (smaller mass) accelerates more than Cart Y when the same force is applied, following the principle that a = F_net/m. Choice A incorrectly claims the more massive cart accelerates more, which contradicts Newton's Second Law—larger mass means smaller acceleration for the same force; Choice C incorrectly suggests both accelerations are equal, ignoring the effect of different masses; Choice D incorrectly suggests 10 N isn't enough to cause acceleration, when any net force causes acceleration regardless of magnitude. This principle explains everyday observations: why it's easier to push an empty shopping cart than a full one, why sports cars (lower mass) can accelerate faster than trucks with the same engine force, and why reducing spacecraft mass is crucial for achieving desired accelerations with limited rocket thrust.
A student applies different net forces to the same cart and records acceleration. Table:
$F_{\text{net}}$ (N) $a$ (m/s $^2$ )

0 0
5 2.5
10 5.0
15 7.5
Which graph description best matches a plot of $a$ (vertical axis) vs. $F_{\text{net}}$ (horizontal axis) for this data?
Answer: A straight line through the origin with positive slope. This question tests understanding that net force and acceleration are directly proportional, as described by Newton's Second Law: F_net = m a. The data show a direct proportional relationship between net force and acceleration: acceleration increases linearly with net force, passing through origin (0 N → 0 m/s²), with constant slope—this means graphing a vs F_net yields a straight line through the origin with positive slope = 1/m. Looking at the data: points (0,0), (5,2.5), (10,5.0), (15,7.5) form a straight line with slope 2.5/5=0.5 (or 1/m, so m=2 kg), confirming the linear proportionality without curvature. Choice A is correct because it properly recognizes the linear pattern in the data: a straight line through the origin with positive slope, matching the plotted points. Choice B is wrong because it claims a curved line (exponential growth), but the data show constant ratio F_net/a=2, not increasing ratios that would indicate non-linearity. Collecting and analyzing force-motion data: (1) set up investigation with constant mass object, (2) apply different net forces (use force sensor or known applied force minus friction), (3) measure resulting acceleration for each force (motion sensor, or calculate from distance and time), (4) record data in table: F_net | a values, (5) graph data: F_net on x-axis, a on y-axis, (6) analyze pattern: should see straight line through origin showing proportionality, (7) calculate slope: slope = Δa/ΔF_net = 1/m, allows determining object's mass from the data.
A $2\,\text{kg}$ cart is pushed along a level track. The net force is changed while the cart's acceleration is measured. Table: $F_{\text{net}}$ (N) vs. $a$ (m/s $^2$ ) - 5 N → 2.5 m/s $^2$ - 10 N → 5.0 m/s $^2$ - 15 N → 7.5 m/s $^2$ Based on the pattern in the data, what acceleration would you predict for $F_{\text{net}}=20\,\text{N}$ on the same cart?
Answer: $10\,\text{m/s}^2$ This question tests understanding that net force and acceleration are directly proportional, as described by Newton's Second Law: F_net = m a, allowing prediction of acceleration for new forces based on the pattern. The data show a direct proportional relationship between net force and acceleration: when net force doubles from 5 N to 10 N, acceleration doubles from 2.5 to 5.0 m/s², and from 5 N to 15 N (tripled), acceleration triples to 7.5 m/s²—this linear pattern means a = F_net / m, with m constant. Looking at the data: the ratio F_net/a is constant at 5/2.5=2, 10/5=2, 15/7.5=2, so m=2 kg; for 20 N, a = 20 / 2 = 10 m/s², following the same proportionality. Choice A is correct because it properly predicts that increasing net force to 20 N (quadrupled from 5 N) will quadruple the acceleration to 10 m/s², consistent with the linear pattern in the table. Choice B is wrong because it suggests 8 m/s², which would not fit the constant ratio (20/8=2.5, but data show 2); it miscalculates the pattern, perhaps confusing with a different mass or non-linear relationship. Collecting and analyzing force-motion data: (1) set up investigation with constant mass object, (2) apply different net forces (use force sensor or known applied force minus friction), (3) measure resulting acceleration for each force (motion sensor, or calculate from distance and time), (4) record data in table: F_net | a values, (5) graph data: F_net on x-axis, a on y-axis, (6) analyze pattern: should see straight line through origin showing proportionality, (7) calculate slope: slope = Δa/ΔF_net = 1/m, allows determining object's mass from the data and predicting for new forces.
A student applies a constant net force to the same cart for 2 seconds each time and measures the change in speed $\Delta v$ . $F_{net}$ (N) | $\Delta v$ after 2 s (m/s) 0 | 0 5 | 5 10 | 10 15 | 15 What does the data show about how net force affects the cart's motion (for the same time interval)?
Answer: Larger net force causes a larger change in speed; $\Delta v$ increases proportionally with $F_{net}$ . This question tests understanding that net force and acceleration are directly proportional, as described by Newton's Second Law: F_net = ma. The data show a direct proportional relationship between net force and acceleration: when net force increases, acceleration increases by the same factor (double the force → double the acceleration), and when net force decreases, acceleration decreases proportionally—this relationship is linear, meaning if you graph F_net (x-axis) vs acceleration (y-axis), you get a straight line through the origin, and the slope of that line equals 1/m (the reciprocal of the object's mass). This proportionality is Newton's Second Law: F_net = ma, which can be rearranged to a = F_net/m, showing that for constant mass, acceleration is directly proportional to net force. Looking at the data: when net force is 5 N, Δv is 5 m/s, when net force increases to 10 N (doubled), Δv increases to 10 m/s (also doubled), and when net force increases further to 15 N (tripled from original), Δv is 15 m/s (also tripled)—this consistent doubling and tripling demonstrates perfect proportionality (since Δv = a*t and t is constant, Δv ∝ a ∝ F_net). The ratio F_net/Δv is constant throughout: 5/5 = 1, 10/10 = 1, 15/15 = 1 (related to m/t, since Δv = (F_net/m)*t, so constant ratio confirms proportionality), confirming Newton's Second Law F_net = ma holds for all data points. The straight line through the origin shows the proportional relationship clearly—the graph passes through (0,0) because zero net force produces zero acceleration (balanced forces), and increases linearly showing each additional Newton of force produces a consistent amount of additional acceleration. Choice A is correct because it accurately identifies the proportional relationship: F_net ∝ a ∝ Δv (for constant time) / correctly interprets the data showing that doubling force doubles Δv / properly recognizes the linear pattern in graph or constant ratio in table / correctly predicts that increasing net force will proportionally increase Δv. Choice B is wrong because it claims an inverse relationship (larger force → smaller Δv) when the data clearly show the opposite: as force increases from 5 N to 10 N to 15 N, Δv increases from 5 to 10 to 15 m/s (both increase together) / suggests there's no relationship or that Δv is independent of force, when the data clearly show Δv changes systematically with net force. Collecting and analyzing force-motion data: (1) set up investigation with constant mass object, (2) apply different net forces (use force sensor or known applied force minus friction), (3) measure resulting acceleration for each force (motion sensor, or calculate from distance and time), (4) record data in table: F_net | a values, (5) graph data: F_net on x-axis, a on y-axis, (6) analyze pattern: should see straight line through origin showing proportionality, (7) calculate slope: slope = Δa/ΔF_net = 1/m, allows determining object's mass from the data. Real investigations might show slight deviations from perfect line (measurement uncertainty, friction variations), but overall pattern should be clear: larger net forces produce larger accelerations proportionally—this relationship F_net = ma is one of the most fundamental in physics, describing how forces cause motion changes in everything from tiny molecules to planets, and collecting data to verify it experimentally (as students can do with carts, ramps, and force sensors) demonstrates how physics laws are not just theoretical but are actually observed patterns in nature that we can measure and test.
A cart of constant mass is tested with different net forces. The results are shown. $F_{net}$ (N) | $a$ (m/s $^2$ ) 0 | 0 4 | 2 8 | 4 12 | 6 Which conclusion best matches the data?
Answer: Acceleration increases by the same amount each time force increases by the same amount (direct proportionality). This question tests understanding that net force and acceleration are directly proportional, as described by Newton's Second Law: F_net = ma. The data show a direct proportional relationship between net force and acceleration: when net force increases, acceleration increases by the same factor (double the force → double the acceleration), and when net force decreases, acceleration decreases proportionally—this relationship is linear, meaning if you graph F_net (x-axis) vs acceleration (y-axis), you get a straight line through the origin, and the slope of that line equals 1/m (the reciprocal of the object's mass). This proportionality is Newton's Second Law: F_net = ma, which can be rearranged to a = F_net/m, showing that for constant mass, acceleration is directly proportional to net force. Looking at the data: when net force is 4 N, acceleration is 2 m/s², when net force increases to 8 N (doubled), acceleration increases to 4 m/s² (also doubled), and when net force increases further to 12 N (tripled from original), acceleration is 6 m/s² (also tripled)—this consistent doubling and tripling demonstrates perfect proportionality. The ratio F_net/a is constant throughout: 4/2 = 2, 8/4 = 2, 12/6 = 2 (this constant ratio equals the object's mass m = 2 kg in this example), confirming Newton's Second Law F_net = ma holds for all data points. The straight line through the origin shows the proportional relationship clearly—the graph passes through (0,0) because zero net force produces zero acceleration (balanced forces), and increases linearly showing each additional Newton of force produces a consistent amount of additional acceleration. Choice A is correct because it accurately identifies the proportional relationship: F_net ∝ a or F_net = ma / correctly interprets the data showing that doubling force doubles acceleration / properly recognizes the linear pattern in graph or constant ratio in table / correctly predicts that increasing net force will proportionally increase acceleration. Choice B is wrong because it claims the relationship is non-linear (curved, exponential, quadratic) when the graph is a straight line and data show constant ratio / doesn't recognize that the constant F_net/a ratio equals mass (fundamental to F = ma). Collecting and analyzing force-motion data: (1) set up investigation with constant mass object, (2) apply different net forces (use force sensor or known applied force minus friction), (3) measure resulting acceleration for each force (motion sensor, or calculate from distance and time), (4) record data in table: F_net | a values, (5) graph data: F_net on x-axis, a on y-axis, (6) analyze pattern: should see straight line through origin showing proportionality, (7) calculate slope: slope = Δa/ΔF_net = 1/m, allows determining object's mass from the data. Real investigations might show slight deviations from perfect line (measurement uncertainty, friction variations), but overall pattern should be clear: larger net forces produce larger accelerations proportionally—this relationship F_net = ma is one of the most fundamental in physics, describing how forces cause motion changes in everything from tiny molecules to planets, and collecting data to verify it experimentally (as students can do with carts, ramps, and force sensors) demonstrates how physics laws are not just theoretical but are actually observed patterns in nature that we can measure and test.
A $2\,\text{kg}$ cart starts from rest. For each trial, a constant net force is applied for $2\,\text{s}$ . Use $\Delta v = a\Delta t$ and $a = F_{\text{net}}/m$ . What is the cart's change in speed after $2\,\text{s}$ when $F_{\text{net}} = 10\,\text{N}$ ?
Answer: $10\,\text{m/s}$ This question tests understanding that net force and acceleration are directly proportional (F_net = ma) and how to calculate velocity change using Δv = aΔt. First, we find acceleration using a = F_net/m = 10 N / 2 kg = 5 m/s²—this tells us how quickly the cart's velocity changes each second. With constant acceleration for 2 seconds, the change in speed is Δv = aΔt = 5 m/s² × 2 s = 10 m/s, meaning the cart goes from 0 m/s (at rest) to 10 m/s after 2 seconds. Choice B is correct because it gives 10 m/s, which results from proper application of both Newton's Second Law (a = F_net/m = 10/2 = 5 m/s²) and the kinematic equation (Δv = aΔt = 5 × 2 = 10 m/s). Choice A (5 m/s) incorrectly uses only 1 second or forgets to multiply by time; Choice C (20 m/s) might come from incorrectly multiplying force by time (10 × 2) without dividing by mass; Choice D (2.5 m/s) appears to divide force by mass×time (10/4) which has no physical meaning. This calculation demonstrates how forces cause velocity changes: (1) net force determines acceleration through F = ma, (2) acceleration tells us the rate of velocity change, (3) multiplying by time gives total velocity change, (4) starting from rest means final velocity equals the change—this process explains everything from car acceleration to rocket launches.
Two students do a tug-of-war with a rope attached to a low-friction cart. They measure the net force on the cart and its acceleration. Trial 1: $F_{\text{net}} = 4\,\text{N}$ , $a = 2\,\text{m/s}^2$ Trial 2: $F_{\text{net}} = 8\,\text{N}$ , $a = 4\,\text{m/s}^2$ What pattern do these results show?
Answer: Acceleration is proportional to net force; doubling $F_{\text{net}}$ doubles $a$ . This question tests understanding that net force and acceleration are directly proportional, as shown by Newton's Second Law: F_net = ma. The data show that when net force doubles from 4 N to 8 N, acceleration doubles from 2 m/s² to 4 m/s²—this perfect doubling demonstrates direct proportionality where F_net ∝ a. The ratio F_net/a remains constant: 4/2 = 2 and 8/4 = 2, indicating the cart's mass is 2 kg (since F_net/a = m), which confirms Newton's Second Law holds for both trials. Choice A is correct because it accurately identifies the proportional relationship: doubling F_net doubles a, which is exactly what happens when force increases from 4 to 8 N and acceleration increases from 2 to 4 m/s². Choice B incorrectly claims inverse proportionality (doubling force would halve acceleration) when both quantities clearly increase together; Choice C wrongly states no relationship exists when the doubling pattern is obvious; Choice D incorrectly suggests a threshold effect when the cart accelerates proportionally at all force values including 4 N. Real-world tug-of-war demonstrates Newton's laws: (1) net force is the difference between opposing pulls, (2) this net force causes acceleration according to F = ma, (3) doubling the net force (stronger team pulling harder) doubles the acceleration, (4) the cart's motion depends only on net force, not individual forces—this shows how vector addition of forces and Newton's Second Law explain everyday motion.
A cart's mass stays constant. A student finds that a net force of 6 N produces an acceleration of 3 m/s $^2$ . If the student increases the net force to 18 N, what acceleration should the student predict based on the pattern $F_{net} \propto a$ ?
Answer: 9 m/s $^2$ This question tests understanding that net force and acceleration are directly proportional, as described by Newton's Second Law: F_net = ma. The data show a direct proportional relationship between net force and acceleration: when net force increases by a certain factor, acceleration increases by the same factor—this relationship is linear, meaning the ratio F_net/a is constant for a given mass. From the given data, we can find this constant ratio: F_net/a = 6 N / 3 m/s² = 2 kg (this is the cart's mass), confirming Newton's Second Law F_net = ma where m = 2 kg. Looking at the prediction: when net force increases from 6 N to 18 N (tripled), acceleration must also triple from 3 m/s² to 9 m/s²—we can verify this using F_net = ma: 18 N = (2 kg)(a), so a = 18/2 = 9 m/s². The proportional relationship means that tripling the force triples the acceleration, maintaining the constant ratio F_net/a = 18/9 = 2 kg. Choice B is correct because it accurately predicts that tripling the force (6 N → 18 N) triples the acceleration (3 m/s² → 9 m/s²). Choice A (6 m/s²) would only be doubling the acceleration when force is tripled; Choice C (3 m/s²) incorrectly suggests acceleration stays the same when force changes; Choice D (18 m/s²) incorrectly assumes a = F_net without considering mass, giving an acceleration value equal to the force value. This proportional relationship F_net ∝ a is fundamental to physics and can be verified experimentally by applying known forces to a constant-mass object and measuring the resulting accelerations—the data will always show this direct proportional pattern.
Two teams play tug-of-war on a smooth surface so friction is small. The net force on the rope is the difference between the teams' pulls. In which trial will the rope's acceleration be greatest (largest change in motion)?
Answer: Team 1 pulls 350 N and Team 2 pulls 300 N (net force 50 N). This question tests understanding that net force determines acceleration, as described by Newton's Second Law: F_net = ma. The net force in tug-of-war is the difference between the two teams' pulls: if Team 1 pulls harder, the net force points toward Team 1, and the magnitude equals |Force₁ - Force₂|—larger net forces produce larger accelerations. For constant mass (the rope), acceleration is directly proportional to net force, so the trial with the largest net force will have the greatest acceleration. Looking at each trial's net force: Trial A has 300 N - 300 N = 0 N net force (balanced forces, no acceleration); Trial B has 320 N - 300 N = 20 N net force; Trial C has 350 N - 300 N = 50 N net force; Trial D has 310 N - 300 N = 10 N net force. Since acceleration is proportional to net force (a = F_net/m), Trial C with 50 N net force will produce the greatest acceleration—this is 2.5 times the acceleration of Trial B (20 N) and 5 times the acceleration of Trial D (10 N). Choice C is correct because it has the largest net force (50 N), which produces the greatest acceleration and largest change in motion. Choice A has zero net force (balanced forces), so there's no acceleration at all—the rope doesn't move; Choice B has 20 N net force, less than half of Trial C's force; Choice D has only 10 N net force, one-fifth of Trial C's force. This demonstrates a key principle: in any situation with opposing forces (tug-of-war, friction vs applied force, etc.), it's the net force that determines acceleration—the winning team isn't necessarily the one pulling hardest, but the one creating the largest difference between the opposing forces.

$F_{\text{net}}$ (N)	$a$ (m/s $^2$ )
0	0
5	2.5
10	5.0
15	7.5

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