This is a problem from K & K, but I changed it very slightly.
A light plane weighing 2,500 lb makes an emergency landing on a short runway. With its engine off, it lands on the runway at 120 ft/s. A hook on the plane snags a cable attached to a 250 lb sandbag and drags the sandbag along. If the coefficient of friction between the sandbag and the runway is 0.4, and if the plane's brakes give an additional retarding force of 300 lb, how long does it take for the plane to come to a stop?
f = force of friction = mu * m * g
viscous force = -C * v, where C is some constant
F = ma
The Attempt at a Solution
Let's say that the positive x is positive direction. The acceleration of the sandbag and the plane is the same since they are connected by a cable. The force of friction is given by
f = mu * m_sandbag * g = - 3200 lb*ft/s^2
And the additional retarding force from the brakes is
f_v = - 300 v(t) lb*ft/s
So we have
- 3200 - 300 v(t) = m_total * a
-3200 - 300 v(t) = 2750 * dv/dt.
After I solved this differential equation and set v(t) = 0, I found that the time required for the plane to come to a stop is t = 23 seconds.
Does this look right? Thank you!