Example of Mechanics Free Response Problem

AP Physics C Practice






An object of mass m is launched from a stationary helicopter towards the Earth with the speed v0. 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, v0, m, g and k.

d) The acceleration of the object before it reaches its terminal velocity is:
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.
ap, physics, mechanics, function, speed

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 - kvt
mg = kvt
vt = mg/k.

c) a = dv/dt
m(dv/dt) = mg - kv
ap, physics, mechanics, function, speed

ap, physics, mechanics, function, speed

ln(v - mg/k) - ln(v0 - mg/k) = -(k/m)t
v = mg/k + (v0 - mg/k)e-(k/m)t

d) a = dv/dt
a = (-k/m)(v0 - mg/k)e-(k/m)t
a = (g - v0k/m)e-(k/m)t
The acceleration can be positive, negative or zero.
If v0 < vt = mg/k, the acceleration is positive,
If v0 > vt = mg/k, the acceleration is positive,
If v0 = vt = mg/k, the acceleration is zero.
4. is the correct answer

e) If v0 < vt = mg/k, the speed is plotted below,
ap, physics, mechanics, function, speed

If v0 > vt = mg/k, the speed is plotted below,
ap, physics, mechanics, function, speed

If the object is launched with v0 = vt = mg/k, the speed is constant and represented by a horizontal line.


AP Physics C Practice