AP Physics 1 1.5: Two-Dimensional Motion

Analyze two-dimensional motion by separating horizontal and vertical components, with projectile motion as the key case of independent component behavior.

[Maximum number: 7]

(7 points, suggested time 13 minutes)

A stunt cyclist builds a ramp that will allow the cyclist to coast down the ramp and jump over several parked cars, as shown above. To test the ramp, the cyclist starts from rest at the top of the ramp, then leaves the ramp, jumps over six cars, and lands on a second ramp.
$H_{0}$ is the vertical distance between the top of the first ramp and the launch point.
$\theta_{0}$ is the angle of the ramp at the launch point from the horizontal.
$X_{0}$ is the horizontal distance traveled while the cyclist and bicycle are in the air.
$m_{0}$ is the combined mass of the stunt cyclist and bicycle.

(a)

Derive an expression for the distance $X_{0}$ in terms of $H_{0}, \theta_{0}, m_{0}$ , and physical constants, as appropriate.

[ 3 ]

For using conservation of energy to find the speed v of the bicycle as it leaves the ramp

1 point

For using kinematics, vertical components, attempting to find the time the bicycle is in the air

1 point

For a correct expression for $X_{0}$ in terms of given quantities

1 point

Example response for part (a)

$\begin{aligned} E_{\text {top }}=E_{\text {bottom }} m_{0} g H_{0}=\frac{1}{2} m_{0} v^{2} v=\sqrt{2 g H_{0}} v_{f y}=v_{i y}+a t -v \sin \theta=v \sin \theta-g t -2 v \sin \theta=-g t \end{aligned}$

Scoring Note:
Using the range equation to get $X_{0}=2 H_{0} \sin 2 \theta_{0}$ is sufficient to earn the second and third points.

Total for part (a) 3 points

(b)

If the vertical distance between the top of the first ramp and the launch point were $2 H_{0}$ instead of $H_{0}$ , with no other changes to the first ramp, what is the maximum number of cars that the stunt cyclist could jump over? Justify your answer, using the expression you derived in part (a).

[ 2 ]

(c)

On the axes below, sketch a graph of the vertical component of the stunt cyclist's velocity as a function of time from immediately after the cyclist leaves the ramp to immediately before the cyclist lands on the second ramp. On the vertical axis, clearly indicate the initial and final vertical velocity components in terms of $H_{0}, \theta_{0}$ , $m_{0}$ , and physical constants, as appropriate. Take the positive direction to be upward.

Figure for Question (c) — AP Physics 1: Algebra-Based

[ 2 ]

For a linear graph with a constant negative slope

1 point

For a graph that starts at $v_{y}$ and ends at $-v_{y}$ , using only allowed variables

1 point

Example response for part (c)

Figure for Answer to Question (c) — AP Physics 1: Algebra-Based

Total for part (c) for Question 17 points

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1.5.B—Describe the motion of an object moving in two dimensions question 1

Question (a)

Question (b)

Question (c)