AP Calculus AB 4.2: Motion and Rates

Practice AP Calculus questions on analyzing rectilinear motion using position, velocity, acceleration, direction, speed, and units.

[Maximum number: 4]

Two particles, H and J, are moving along the x-axis. For $0 \leq t \leq 5$ , the position of particle H at

time t is given by $x_{H}(t)=e^{t^{2}-4 t}$ and the velocity of particle J at time t is given by

$v_{J}(t)=2 t\left(t^{2}-1\right)^{3}$ .

(a)

During what open intervals of time t, for 0<t<5, are particles H and J moving in opposite

directions? Give a reason for your answer.

[ 3 ]

B During what open intervals of time t, for 0<t<5, are particles H and J moving in opposite directions?

Give a reason for your answer.

From part A, $x_{H}{ }^{\prime}(t)=v_{H}(t)=(2 t-4) e^{t^{2}-4 t}$ . $\begin{aligned} & x_{H}^{\prime}(t)=(2 t-4) e^{t^{2}-4 t}=0 \Rightarrow t=2 & x_{H}^{\prime}(t)<0 \text { for } 0<t<2, \text { and } x_{H}^{\prime}(t)>0 \text { for } 2<t<5. \end{aligned}$	Considers sign of $x_{H}{ }^{\prime}(t)$ or $v_{J}(t)$	Point 3 (P3)
	Analysis for one particle	Point 4 (P4)
Thus, particle H is moving to the left for 0<t<2 and moving to the right for 2<t<5. $\begin{aligned} & v_{J}(t)=2 t\left(t^{2}-1\right)^{3}=0 \text { for } 0<t<5 \Rightarrow t=1<br>& v_{J}(t)<0 \text { for } 0<t<1, \text { and } v_{J}(t)>0 \text { for } 1<t<5. \end{aligned}$ <br>Thus, particle J is moving to the left for 0<t<1 and moving to the right for 1<t<5.<br>Therefore, particles H and J are moving in opposite directions for 1<t<2.	Answer with reason	Point 5 (P5)
Scoring Notes for Part B

- To earn P3, a response can do one of the following:

○ Set $x_{H}{ }^{\prime}(t)=0, v_{H}(t)=0$ , or $(2 t-4) e^{t^{2}-4 t}=0$

○ Set $v_{J}(t)=0$ or $2 t\left(t^{2}-1\right)^{3}=0$

○ Identify t=2 for particle H and no other values in the interval 0<t<5

○ Identify t=1 for particle J and no other values in the interval 0<t<5

○ Identify the interval 1<t<2

- To earn P4, a response can provide an analysis of signs of velocity or direction of motion on the

interval 0<t<5 for either particle H or particle J.

- To be eligible for P5, a response must provide correct analyses of signs of velocity or direction of

motion on the interval 0<t<5 for both particles.

- Only analysis within the interval 0<t<5 will be considered in scoring.

(b)

It can be shown that $v_{J}{ }^{\prime}(2)>0$ . Is the speed of particle J increasing, decreasing, or neither at

time t=2 ? Give a reason for your answer.

[ 1 ]

C It can be shown that $v_{J}^{\prime}(2)>0$ . Is the speed of particle J increasing, decreasing, or neither at time

t=2 ? Give a reason for your answer.

$v_{J}(2)>0$ and $v_{J}^{\prime}(2)>0$ .

Answer with reason

Point 6 (P6)

Because $v_{J}(2)$ and $v_{J}{ }^{\prime}(2)$ have the same sign, the speed of

particle J is increasing at t=2.

Scoring Notes for Part C

- An evaluation of $v_{J}(2)$ is not necessary, but if a value is presented, it must be correct. The correct

value is $v_{J}(2)=108$ .

- An evaluation of $v_{J}{ }^{\prime}(2)$ is not necessary, but if a value is presented, it must be correct. The correct

value is $v_{J}{ }^{\prime}(2)=486$ .

- A response can either import the analysis for the sign of $v_{J}(2)$ from part B or restart.

- A response that stated " $v_{J}(t)>0$ for 1<t<5 " in part B does not need to restate $v_{J}(2)>0$ and

earns P6 for " $v_{J}(2)$ and $v_{J}{ }^{\prime}(2)$ have the same sign, so the speed is increasing."

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CHA-3.B—Calculate rates of change in applied contexts question 1

Question (a)

Question (b)