Uniform Circular Motion: Speed of the bullet

[mehacute]

Homework Statement

Derive a formula for the bullet speed v in terms of D, T, and a measured angle theta between the position of the hole in the first disk and that of the hole in the second. If required, use pi, not its numeric equivalent. Both of the holes lie at the same radial distance from the shaft. theta measures the angular displacement between the two holes; for instance, \theta = 0 means that the holes are in a line and \theta=\pi means that when one hole is up, the other is down. Assume that the bullet must travel through the set of disks within a single revolution.

Homework Equations

So, I know that speed = 2*pi*r/t and that speed = r*omega. but I don't know how to factor in the theta. Can someone please derive the whole thing and explain? Thank you.

The Attempt at a Solution

 

[Doc Al]

You didn't describe the problem completely, but I assume you're dealing with a two-disk velocity selector? The disks are moving at some angular speed omega?

If so, here's a hint: How much time does it take for a disk to move through the angle theta?

 

[tomcue]

To derive the formula for the bullet speed, we can use the formula for angular velocity, which is given by omega = theta/t, where theta is the angular displacement and t is the time taken for the displacement. Since the bullet must travel through the set of disks within a single revolution, the time taken for the displacement can be written as T = 2*pi/omega.

Next, we can use the formula for linear velocity, which is given by v = r*omega, where r is the radial distance from the shaft. Since both holes lie at the same radial distance, we can write r as D/2, where D is the diameter of the disks.

Substituting the value of omega from the first equation into the second equation, we get:

v = (D/2)*(theta/T)

Since T = 2*pi/omega, we can rewrite the equation as:

v = (D/2)*(theta/(2*pi/omega))

Simplifying further, we get:

v = (D*omega*theta)/(4*pi)

Substituting the value of omega = theta/t, we get:

v = (D*theta^2)/(4*pi*t)

Therefore, the formula for the bullet speed in terms of D, T, and theta is:

v = (D*theta^2)/(4*pi*t)

I hope this helps to derive the formula and explain it. Please let me know if you have any further questions.