Draw the initial velocity components and the acceleration vector on a simplified sketch of the rock immediately after launch. Label the relevant quantities.

Derive an expression for the positive time $t_f$ at which the rock reaches a point that is a distance $h$ below the launch point. Express your answer in terms of $v_0$, $\theta$, $h$, and $g$.

Calculate the rock’s time of flight for the situation shown.

Calculate the horizontal displacement of the rock.

Calculate the magnitude and direction of the rock’s velocity immediately before impact.

Determine whether the new impact speed is less than, equal to, or greater than the impact speed calculated in part (c)(iii).

Justify your answer using the horizontal and vertical components of velocity.

Draw the velocity vector at one of the ball positions on the ascending portion of an arc. Resolve the vector into labelled horizontal and vertical components.

Sketch a graph of vertical velocity $v_y$ as a function of time $t$ from immediately after one bounce until immediately before the next bounce. Indicate the instant at which the ball reaches its maximum height.

Sketch the corresponding graph of vertical position $y$ as a function of time $t$. Show how the graph is consistent with your answer to part (b)(i).

Let $f$ be the flash frequency and let $\Delta x$ be the horizontal distance between successive recorded positions.

Derive an expression for $\Delta x$ in terms of the horizontal velocity $v_x$ and $f$.

Determine the time interval between successive recorded positions.

Justify why the horizontal spacing is approximately constant while the vertical spacing changes throughout an arc.

Determine how the horizontal spacing and the number of recorded positions during one arc change.

Justify whether the physical trajectory of the ball changes.

Indicate the independent variable, the dependent variable, and the measurements needed to obtain each variable.

Describe a repeatable experimental procedure. Include how the photogate is used and identify important variables that must be controlled.

Describe a graph that can be used to determine the cart’s acceleration and test whether the acceleration is constant.

Plot $v^2$ on the vertical axis as a function of $2x$ on the horizontal axis. Label both axes with units, use sensible linear scales, plot all six data points, and draw a best-fit line.

Determine the cart’s acceleration from the graph.

Determine the cart’s initial speed at $x=0$.

Verify whether the data support a constant-acceleration model.

Determine the direction of the ball’s velocity at the highest point of its trajectory.

Justify why the velocity has the direction given in part (a) even though the acceleration is nonzero at that point.

Derive an equation for the ball’s vertical position $y$ as a function of its horizontal position $x$. Express your answer in terms of $x$, $v_0$, $\theta$, and $g$.

Determine whether the second ball is above, below, or at the same vertical position as the first ball when both have reached the same horizontal position $x>0$.

Justify your answer using the equation derived in part (c).

Determine the object's acceleration during $0\le t\le4.0\,\mathrm{s}$.

Calculate the object's displacement during $0\le t\le7.0\,\mathrm{s}$.

Determine the time at which the object reverses direction during $7.0\le t\le11.0\,\mathrm{s}$.

Derive an expression for the object's position $x(t)$ during $7.0\le t\le11.0\,\mathrm{s}$. Your expression must use the graph and the given initial position.

Calculate the object's position at $t=11.0\,\mathrm{s}$.

Compare the object's average speed with the magnitude of its average velocity during $0\le t\le11.0\,\mathrm{s}$.

Justify your comparison using evidence from the graph.

Determine the object's average velocity during $0\le t\le1.0\,\mathrm{s}$.

Estimate the object's instantaneous velocity at $t=2.0\,\mathrm{s}$ by using graph values on either side of that time.

Draw a motion diagram for the object at $t=0$, $1.0$, $2.0$, $3.0$, and $4.0\,\mathrm{s}$. Show position dots, velocity arrows, and acceleration arrows.

Derive the object's initial velocity $v_0$ and acceleration $a_x$ by using the graph values at $t=0$, $1.0$, and $2.0\,\mathrm{s}$ with a constant-acceleration relationship.

Sketch the corresponding graph of velocity $v_x$ as a function of time $t$ from $t=0$ to $t=4.0\,\mathrm{s}$. Include numerical intercepts and the zero crossing.

Justify why the maximum of the provided position graph is consistent with your velocity graph.

Determine the time at which the object would first stop if its initial velocity were doubled while its acceleration and initial position remained unchanged.

Determine the ruler distance that corresponds to a sensor reading of $d_s=1.02\,\mathrm{m}$.

Describe a feasible procedure that uses the calibrated motion sensor to determine whether a cart moving along a straight track has constant acceleration and to measure that acceleration.

Describe important controls and repeated measurements that would improve the investigation.

Describe the graph that should be produced from the measured motion data and how the acceleration would be obtained from that graph.

Calculate the value of $t^2$ for Trial $4$.

Plot $x$ on the vertical axis as a function of $t^2$ on the horizontal axis. Label both axes with units, use a sensible scale, plot all six points, and draw a best-fit line.

Determine the cart's acceleration from the slope of the best-fit line.

Justify whether the data are consistent with constant acceleration.

Compare the increase in stopping distance from $v_0=5\,\mathrm{m\,s^{-1}}$ to $10\,\mathrm{m\,s^{-1}}$ with the increase from $10\,\mathrm{m\,s^{-1}}$ to $15\,\mathrm{m\,s^{-1}}$.

Justify the curved shape of the graph by applying a constant-acceleration kinematic relationship.

Derive an expression for $d$ in terms of $v_0$ and $b$.

Calculate the acceleration magnitude $b$ by using the point on the graph at $v_0=20\,\mathrm{m\,s^{-1}}$.

Determine the factor by which $d$ changes when $v_0$ is multiplied by $1.5$.

Verify the statement that doubling the initial speed requires four times the stopping distance under the stated conditions.

Determine the projectile's vertical acceleration.

Determine the time at which the projectile reaches its highest point.

Calculate the projectile's vertical displacement from launch to impact by using signed areas on the graph.

Derive an expression for the maximum vertical displacement above the launch point in terms of the initial vertical speed $v_{y0}$ and the acceleration magnitude $g$. Use the geometry of the graph.

Calculate the maximum vertical displacement above the launch point.

Calculate the horizontal range from launch to impact.

Calculate the magnitude of the projectile's velocity immediately before impact.

Justify whether the magnitude of the projectile's acceleration changes at the highest point.

Determine the velocity $v_{PG}$ of object $P$ relative to the ground.

Determine the velocity $v_{QG}$ of observer $Q$ relative to the ground.

Determine the time at which $P$ and $Q$ meet.

Determine their position in the ground frame when they meet.

Draw a one-dimensional vector diagram that represents the relationship among $v_{PG}$, $v_{QG}$, and the velocity $v_{PQ}$ of $P$ relative to $Q$.

Derive an expression for the position $x_{PQ}(t)$ of $P$ relative to $Q$ by using the intercepts and slopes of the provided graph.

Sketch a graph of $x_{PQ}$ as a function of $t$ from $t=0$ to $t=6.0\,\mathrm{s}$. Include the vertical intercept, zero crossing, and value at $t=6.0\,\mathrm{s}$.

Compare the acceleration of $P$ measured in frame $G$ with the acceleration of $P$ measured in the inertial frame of $Q$.

Justify whether $Q$ observes $P$ moving to the right or to the left before they meet.