Proportional Resistive Force Problem

[Nitrus]

Consider an object which the net force is a resistive force proportional to the square of its speed. For example: assume that the resistive force acting on a speed skater is f=-kmv^2, where k is a constant and m is the skater's mass. The skater corsses the finish line of a straight-line race with speed v(i) and the slows down by coasting on his skates. Show that his speed at time "t", any time after the finish line is equal to Vf=Vi/(1+Vi*k*t).

Do i need to do a diff eq for this?

 

[HallsofIvy]

I don't know that you NEED a differential equation- it's a pretty straight forward integration- but it might be simpler to think of it as a differential equation.

The basic law is, of course, "mass times acceleration equals force". Here, you are told that the force is kmv² so

m dv/dt= km v² (yep, that's a differential equation!)
which you can immediately write as

v^-2 dv= k dt (Now it's an integral problem!)

Integrate both sides. Be careful about the constant of integration.

 

[Nitrus]

yeah i figured it out... you take the definite integral of the right side from t to 0, then on the left you do as V-knot goes to V(t). which will give you:

1/V(t)-1/Vi=-kt

which then simplifies to what I am looking for