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AP PHYSICS C: MECHANICS

Unit $2$ Progress Check: FRQ

Show all your work for each part of the question. The parts within the question may not have equal weight.

Figure $1$

A small sphere of mass $m$ is launched straight upward from the top of a high cliff, as shown in Figure $1,$ with an initial speed $v_0$ at time $t=0.$ The air exerts a resistive force $F_r$ on the sphere that varies with velocity $v$ according to the equation vec$(F{)}_r=-$bvec$(v),$ where $b$ is a positive constant. The sphere reaches its maximum height in the air at time $t_h$ and then falls back toward the ground, which is a long distance below the sphere's launch point at the top of the cliff. In the following questions, the positive direction is taken to be upward.

$($a$)$

$($i$)$ Derive an equation for the velocity $v$ of the sphere as a function of time $t$ by solving a differential equation. Express your answer in terms of $b,m,t,v_0,$ and physical constants, as appropriate.

$($ii$)$ On the axes in Figure $2,$ sketch a graph of the sphere's acceleration as a function of $t.$ The sphere does not reach the ground during the time interval shown in the graph. The time $t_h$ and acceleration values of $g$ and $-g$ are indicated on the axes.

AP Physics C: Mechanics

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