# Friction and nonconstant acceleration?

1. May 24, 2004

### Zorodius

A problem in my book reads as follows:

A 1000 kg boat is traveling at 90 km/h when its engine is shut off. The magnitude of the frictional force $\vec{f_k}$ between boat and water is proportional to the speed v of the boat: $f_k = 70v$, where v is in meters per second and $f_k$ is in newtons. Find the time required for the boat to slow to 45 km/h.

My question with this is: g'nhuh? If the magnitude of the frictional force is a function of velocity, that seems to imply that the acceleration is not constant. I was under the impression that the equations for motion and friction that I had been given so far applied only to constant acceleration. I tried to solve this by converting the measurements into meters per second (25 m/s when the engine is shut off, slows to 12.5 m/s) and then guessing that, since a=f/m, then a=70v/1000, and perhaps I could say v = 25 - 70 v / 1000 * t. I solved this for v, and graphically found that v = 12.5 when t is about 14.6. Unfortunately, that wasn't the right answer, which isn't particularly surprising since I'm unsure where to go with this from the very start.

A little help?

2. May 24, 2004

### Staff: Mentor

integrate!

The force and the acceleration are not constant. You can't apply the simple kinematic equations for uniform acceleration to this problem. You have to apply F=ma to set up a simple differential equation. You'll need to integrate! I hope you've covered a little calculus.

$$F = ma = m \frac{dv}{dt}$$
$$70v = m \frac{dv}{dt}$$

Etc...