An object of mass m is launched from a stationary helicopter towards the Earth with the speed v

_{0}. It experiences a force of air resistance F = -kv, where k is a positive constant. The positive direction of all vector quantities is downward.

a) On the dot below draw and label vectors to represent the forces acting on the object.

b) What is the terminal speed of the object?

c) Write and solve a differential equation to find the speed of the object as a function of time, v

_{0}, m, g and k.

d) The acceleration of the object before it reaches its terminal velocity is:

- 1. positive
- 2. negative
- 3. equal to zero
- 4. can be any of the above

Choose one of the four options above and explain your choice.

e) On the axes below plot the speed of the object versus time.

**Solution:**

a) Since velocity is downward, air resistance is upward, in the opposite direction of gravity.

b) When the object reaches terminal velocity, its acceleration is a = 0 and the sum of all forces acting on the object is 0.

0 = mg - kv

_{t}

mg = kv

_{t}

v

_{t}= mg/k.

c) a = dv/dt

m(dv/dt) = mg - kv

ln(v - mg/k) - ln(v

_{0}- mg/k) = -(k/m)t

v = mg/k + (v

_{0}- mg/k)e

^{-(k/m)t}

d) a = dv/dt

a = (-k/m)(v

_{0}- mg/k)e

^{-(k/m)t}

a = (g - v

_{0}k/m)e

^{-(k/m)t}

The acceleration can be positive, negative or zero.

If v

_{0}< v

_{t}= mg/k, the acceleration is positive,

If v

_{0}> v

_{t}= mg/k, the acceleration is positive,

If v

_{0}= v

_{t}= mg/k, the acceleration is zero.

4. is the correct answer

e) If v

_{0}< v

_{t}= mg/k, the speed is plotted below,

If v

_{0}> v

_{t}= mg/k, the speed is plotted below,

If the object is launched with v

_{0}= v

_{t}= mg/k, the speed is constant and represented by a horizontal line.

AP Physics C Practice