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?
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?