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## Homework Statement

You're firing a ball of mass 10.0g out of a spring gun, and the spring is the entire length of the barrel, which is .05m long. The force constant for the spring is 500 N/m. There is a constant resisting force of 10.0N acting on the ball as it travels down the length of the barrel. What is the speed of the ball as it leaves the gun?

## Homework Equations

mx'' + cx' + kx = 0

Work total = ΔK

Work = F*d

K = 1/2mv^2

Potential energy of a spring = 1/2kx^2

## The Attempt at a Solution

So basically, I'm wondering if it's possible to solve this problem using the above differential equation. I know that the cx' term refers to the damping force, which I assume to be the resisting force. So I tried my equation to be:

.01x'' + 10 + 500x = 0

I let time t= 0 correspond to when the spring has been fully compressed, so x(0) = -.05 (I let x = 0 be at the end of the barrel), and the block is not moving when the spring is compressed, so x'(0) = 0.

I then solve the differential equation, and find the time it takes for the spring to reach it's uncompressed distance, when x = 0.

After I find the time taken, I plug that into x'(t) to find the velocity of the block when it leaves the spring. However, the answer I get from doing this doesn't match the answer I get when using conservation of energy techniques.

For conservation of energy, I said that:

Work total = ΔK, where the total work done is due to the resisting force, and the spring's potential energy.

So I had:

(1/2)(500)(.05)^2 - (10)(.05) = 1/2mv^2

solved for v, and got 5 m/s.

So, unless I'm doing something wrong in my energy approach, the answer should be 5m/s.

Basically what I'm asking is, is it possible to model this situation with the mass-spring differential with the information given, and if so, what would be the proper way to handle the cx' term?