Kinematics and Motion Variables (4A) - Chem & Physics

1. Which graph's slope equals instantaneous velocity: $x$ vs $t$ or $v$ vs $t$?

   Slope of an $x$ vs $t$ graph equals instantaneous velocity. The slope on a position-time graph represents the rate of change of position, which defines velocity at that instant.

2. Find $v$ after $t = 3\,\text{s}$ if $v_0 = 2\,\text{m/s}$ and $a = 4\,\text{m/s}^2$.

   $v = 14\,\text{m/s}$. Using $v = v_0 + at$, final velocity increases linearly from initial under constant positive acceleration.

3. Find $a$ if $v_0 = 5\,\text{m/s}$, $v = 1\,\text{m/s}$, and $t = 2\,\text{s}$.

   $a = -2\,\text{m/s}^2$. Acceleration is found from $a = \frac{v - v_0}{t}$, yielding negative value for deceleration in this case.

4. What is the definition of instantaneous velocity using calculus notation?

   Instantaneous velocity is $v = \frac{dx}{dt}$. Instantaneous velocity is the derivative of position with respect to time, indicating velocity at a specific moment.

5. Find $v_{0x}$ and $v_{0y}$ if speed is $10\,\text{m/s}$ at $30^\circ$ above horizontal.

   $v_{0x}=10\cos30^\circ$, $v_{0y}=10\sin30^\circ$. Initial velocity components resolve using trigonometry, with cosine for horizontal and sine for vertical relative to the angle.

6. Find time to reach max height if a ball is thrown upward with $v_0 = 20\,\text{m/s}$ and $a = -g$.

   $t = \frac{v_0}{g} \approx 2.0\,\text{s}$. Time to max height occurs when $v=0$, so $t = \frac{-v_0}{a}$ with $a=-g$, simplifying to $v_0 / g$.

7. State the constant-acceleration equation relating $v^2$, $v_0^2$, $a$, and displacement.

   $v^2 = v_0^2 + 2a(x - x_0)$. This equation eliminates time by combining velocity and position equations, relating speeds squared to acceleration and displacement.

8. Identify the sign of acceleration when an object moves in $+x$ but slows down.

   Acceleration is negative: $a < 0$. When slowing in the positive direction, acceleration opposes motion, resulting in a negative value in the coordinate system.

9. What is the SI unit of acceleration expressed in base units?

   Meters per second squared: $\text{m}\,\text{s}^{-2}$. Acceleration, as change in velocity over time, has SI units derived from meters per second per second.

10. What is the magnitude and direction of gravitational acceleration near Earth's surface?

    $g \approx 9.8\,\text{m}\,\text{s}^{-2}$ downward. Near Earth's surface, gravity causes constant downward acceleration, approximated as 9.8 m/s² for kinematic calculations.

11. Which graph's slope equals instantaneous acceleration: $v$ vs $t$ or $x$ vs $t$?

    Slope of a $v$ vs $t$ graph equals instantaneous acceleration. The slope on a velocity-time graph indicates the rate of change of velocity, corresponding to acceleration at that point.

12. State the kinematic equations for projectile components (no air resistance) in 2D.

    $x=x_0+v_{0x}t$ and $y=y_0+v_{0y}t-\frac{1}{2}gt^2$. Projectile equations separate into horizontal constant velocity and vertical constant acceleration due to gravity.

13. State the constant-acceleration equation for displacement using average of $v_0$ and $v$.

    $x - x_0 = \frac{(v_0 + v)}{2}t$. Displacement equals average velocity times time, where average is the mean of initial and final velocities under constant acceleration.

14. In projectile motion without air resistance, which velocity component changes: $v_x$ or $v_y$?

    $v_y$ changes; $v_x$ is constant. In projectile motion, gravity affects only the vertical component, leaving horizontal velocity unchanged without air resistance.

15. What is the definition of distance traveled in one dimension?

    Distance is total path length traveled (nonnegative scalar). Distance measures the total length of the path taken, always positive as a scalar quantity regardless of direction.

16. State the constant-acceleration equation relating $v$, $v_0$, $a$, and $t$.

    $v = v_0 + at$. This equation derives from constant acceleration, integrating acceleration to find velocity as initial plus acceleration times time.

17. State the constant-acceleration equation relating $x$, $x_0$, $v_0$, $a$, and $t$.

    $x = x_0 + v_0 t + \frac{1}{2}at^2$. This position equation results from integrating velocity under constant acceleration, combining initial terms and quadratic acceleration effect.

18. What is the definition of instantaneous acceleration using calculus notation?

    Instantaneous acceleration is $a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$. Instantaneous acceleration is the derivative of velocity with respect to time, or second derivative of position, showing rate of velocity change.

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