Integrating Force: Derive Distance L for AP Question

[aber]

This last part of an AP questions is giving me some trouble, mostly because i involves integrating and i never took Calculus.

Part D: The dart is now shot into a block of wood that is in a fixed place. The block exerts a Force F on the dart that is proportional to the dart's Velocity V and in the opposite direction, that is F=-bv, where b is a constant. Derive and expression for the distance L that the dart penetrates into the block, in terms of m (mass), v (initial), and b.

Since the Force -bv is not constant, i can't figure out how to use the kinetic energy 1/2mv^2 to solve for distance, i would think you would need X=Xknot+vknotT+1/2aT^2.

 

[Doc Al]

Since the Force -bv is not constant, i can't figure out how to use the kinetic energy 1/2mv^2 to solve for distance, i would think you would need X=Xknot+vknotT+1/2aT^2.

As you seem to realize, you need to be able to integrate to solve this problem. That kinematic equation is only good for uniformly accelerated motion, which is not the case here.

If you want to try your hand at integrating, here's a hint: \int F dt = \Delta (mv).

 

[jdavel]

aber,

Here's another way to think about it. You have the equation F = -vb. What does Newton's 2nd law say about F? Can you rewrite what it says as a derivative of v intstead of x?

 

[aber]

jdavel: i don't know how to derive...
Doc Al: Would it be change in F dt= change in MV, F= MV-M0/t=MV/t? t=2L/3V, F= 3MV^2/2L

 

[OlderDan]

You have learned that acceleration is the rate of change of velocity. Since you know the initial velocity, you know the initial force. That force is going to reduce the velocity. If you break the problem up into small intervals of time, you can get a good approximation of the change in velocity in one interval of time by assuming the intial force is constant for the first interval and calculating the change in velocity resulting from that force applied in that short time. At the end of that time, the dart will have a lower velocity, so you can calculate the reduced force and assume it is constant for the next little interval of time, leading to a lower velocity and a lower force for the next interval, etc., etc. If you know how to use a spreadsheet or program a calculator or computer, you can make the times intervals very small and make the approximation as good as you like.