# AP Mechanics Free Response Problem

1. Apr 9, 2007

### harmonicmotion

1. The problem statement, all variables and given/known data
A ball of mass M is thrown vertically upward with an initial speed Vo. It experiences a force of air resistance given by F=-kv, where k is a positive constant. The positive direction for all vector quantities is upward. Express all algebraic answers in terms of M, k, and Vo, and fundamental constants.
a.) Does the magnitude of the acceleration of the ball increase, decrease, or remain the same
b.) Write, but do not solve, a differential equation for the instantaneous speed v of the ball in terms of time t as the ball moves upward
c.) Determine the terminal speed of the ball as it moves downward
d.) Does it take longer for the ball to rise to its maximum height or fall from its maximum height back to the height from which it was thrown?

2. Relevant equations

3. The attempt at a solution

2. Apr 9, 2007

### harmonicmotion

I believe for part a.) that the acceleration decreases as the force of gravity acts on the ball

3. Apr 9, 2007

### harmonicmotion

can someone help?

4. Apr 9, 2007

### r16

When problems like this get confusing I always go back to the basics. Draw a force diagram of the ball ascending. You have 2 forces, a gravitational force and a drag force:

$$\sum F = -F_g - F_f = -mg -kv$$

There is a caveat, however. The "famous 4" are only used when there is a constant acceleration, however, this net force is not constant but a function of $v$.

Your answer for part a looks right to me, the sum of forces and thus the acceleration decreases as the ball ascends not because of a change in the gravitational force which is constant (both $m$ and $g$ are invariant) but because of a change of the drag force (which varies as $v$).

Hope this starts you off.

Last edited: Apr 10, 2007