Calculating Bullet Speed from Angular Displacement & Revolutions

[quicknote]

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.

The picture can be found here.
http://ca.geocities.com/canbball/MRB_rr_8_a.jpg

I know that v = (delta)D/(delta)T where D is distance and T is the period.
But I'm not sure how to find the ratio between the angle and the measure of a full revolution.
Any help would be much appreciated!

 

[Integral]

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.

The picture can be found here.
http://ca.geocities.com/canbball/MRB_rr_8_a.jpg

I know that v = (delta)D/(delta)T where D is distance and T is the period.
But I'm not sure how to find the ratio between the angle and the measure of a full revolution.
Any help would be much appreciated!

your equation
[tex] v = \frac {\Delta D} {\Delta T} [/tex]
Cannot be correct. Both D and T are constants of your problem, so do not change.

You need to start with

[tex] v= \frac D t [/tex]
where t is the time required to travel the distance between the disks. You are given that it take T secs to do a complete rotation, what is that in terms of Pi?

your goal is to find an expression for t involving T and [itex] \Theta [/itex].

 

[quicknote]

I would approach this problem by first understanding the physical principles involved. In this case, we are dealing with rotational motion and the relationship between angular displacement, revolutions, and speed.

To derive a formula for bullet speed v, we can use the equation v = (delta)D/(delta)T, where (delta)D is the change in distance and (delta)T is the change in time. In this case, the change in distance is the distance traveled by the bullet, which is equal to the circumference of the disk. This can be calculated using the formula C = 2(pi)r, where C is the circumference, pi is the ratio of a circle's circumference to its diameter, and r is the radius of the disk.

Now, we need to find the change in time (delta)T. This can be calculated by dividing the total time taken for the bullet to travel through the set of disks by the number of revolutions it makes. In other words, (delta)T = T/N, where T is the total time and N is the number of revolutions.

Next, we need to find the ratio between the angle and the measure of a full revolution. This can be calculated by dividing the angle theta by 2(pi). This is because 2(pi) radians is equal to a full revolution (360 degrees).

Putting all of this together, we can derive the formula for bullet speed v as follows:

v = (C/T) * (theta/2(pi))

Substituting C = 2(pi)r and (delta)T = T/N, we get:

v = (2(pi)r/T) * (theta/2(pi))

Simplifying, we get:

v = (r/T) * theta

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

v = (D/T) * theta

This formula can be used to calculate the speed of the bullet based on the measured angle theta and the distance D and time T. It is important to note that this formula assumes that the bullet travels through the set of disks within a single revolution.

In conclusion, by understanding the principles of rotational motion and using the appropriate equations, we can derive a formula for calculating bullet speed from angular displacement and revolutions. This formula can be used to accurately determine the speed of a bullet based on experimental data.