# Homework Help: Calculating velocity of a bullet with quadratic air drag

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1. Feb 23, 2015

### Matt Q

So this problem was a 2 part question. The first part goes as such.
1. A gun is fired straight up. Assuming that the air drag on the bullet varies quadratically with speed, show that the speed varies with height according to the equations:

v2 = Ae-2kx - (g/k) (upward motion)

v2 = (g/k) - Be2kx (downward motion)

in which A and B are constants of integration, g is the acceleration of gravity, and k = (c2 /m) where c2 is the drag constant and m is the mass of the bullet.

Now this problem I did not have any particular problem solving after some guidance from my professor. For this one I just took the integral of F(v) = -mg - cv2 and F(v) = mg - cv2 for the upward and downward motions respectively. It is the following question that I am hungup on.

2. Use the above result to show that, when the bullet hits the ground on its return, the speed is equal to the expression:

((v0vt) / ((v02 + vt2)^(1/2))

in which v0 is the initial upward speed and vt = (mg/c2)^(1/2) = terminal speed = (g/k)^(1/2).

I am primarily having the problem of setting up this problem. Since it wants us to take into consideration the upward and downward motion of the bullet, I would assume you would want to add the velocities of the first problem, but I could be entirely wrong about that. Also considered starting out with taking the sum of the forces of the first problem:

F(v) = -mg -cv2 and F(v) = mg-cv2 but I don't think that would work out if I integrated it then. Any help on setting up this problem would be appreciated :)

2. Feb 23, 2015

### TSny

Hi, Matt Q. Welcome to PF!

Can you use your "upward" equation to relate the constant A to the initial and terminal speeds?

Can you use the same equation to relate the maximum height to the initial and terminal speeds?

3. Feb 23, 2015

### BvU

Hello Matt, welcome to PF

Nice exercise. I haven't cracked it yet, but if I were you I'd do what it says: use the above result and see how far you can get.
You have v02 = A - g/k, vt2 = g/k and you want vf2=g/k - B.
If you can connect A and B you should have cracked it. Any idea where to make this link ?

T was 60sec faster, but seems to agree. I leave you two to finish this up -- and go to bed.

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