How Do Rotating Disks Measure Bullet Speed?

[Parzival]

Homework Statement

A device can be used to measure the speed of a bullet. The device consists of two rotating disks, separated by a distance of d = 0.850 m, and rotating with an angular speed of 95.0 rad/s. The bullet first passes through the left disk and then through the right disk. It is found that the angular displacement between the two bullet holes is 0.240 rad. From these data, determine the speed of the bullet.

Homework Equations

Angular displacement = arc length/radius
Angular velocity = angular displacement/time

The Attempt at a Solution

I attempted to set up an equation, but I simply couldn't.

 

[voko]

What is the angular displacement of the first disk during the bullet's flight between the disk? Of the second one?

 

[CWatters]

The definition of Velocity is distance/time

The distance is known so you just need to work out the time it takes to travel the distance between the discs.

The definition of angular velocity is

Angular velocity = angle/time

So rearrange that to give an equation for the time. Plug that into the first equation.

 

[Parzival]

The definition of Velocity is distance/time

The distance is known so you just need to work out the time it takes to travel the distance between the discs.

The definition of angular velocity is

Angular velocity = angle/time

So rearrange that to give an equation for the time. Plug that into the first equation.

Please clarify? I've been stumped for a long time.
Following your advice, time = angle/angular velocity
But how do I plug it in?

 

[Nickie]

First, we can use the equation for angular displacement to find the distance traveled by the bullet on each disk:

θ = s/r

Where θ is the angular displacement, s is the distance traveled, and r is the radius of the disk.

For the left disk, the distance traveled is d/2 since the bullet passes through the center of the disk. So, we have:

0.240 rad = (d/2)/r

For the right disk, the distance traveled is (d/2)+x, where x is the distance between the two bullet holes. So, we have:

0.240 rad = ((d/2)+x)/r

We can solve for x by setting the two equations equal to each other:

(d/2)/r = ((d/2)+x)/r

Solving for x, we get:

x = d/2

Now, we can use the equation for angular velocity to find the speed of the bullet:

ω = θ/t

Where ω is the angular velocity, θ is the angular displacement, and t is the time it takes for the bullet to pass through the two disks.

We know that the angular velocity is 95.0 rad/s and the angular displacement is 0.240 rad. So, we have:

95.0 rad/s = 0.240 rad/t

Solving for t, we get:

t = 0.002526 s

Now, we can use the equation for linear velocity to find the speed of the bullet:

v = d/t

Where v is the linear velocity, d is the distance traveled, and t is the time it takes for the bullet to pass through the two disks.

We know that the distance traveled is d, which is given as 0.850 m, and the time is 0.002526 s. So, we have:

v = 0.850 m/0.002526 s

Solving for v, we get:

v = 336.1 m/s

Therefore, the speed of the bullet is approximately 336.1 m/s.