Ballistic Pendulum speed of a bullet

In summary, the speed of a rifle bullet can be measured using a ballistic pendulum. By applying the conservation of energy equation and using the maximum height of the swinging wooden block, the speed of the bullet can be expressed as Vb=[(M+m)/m]x(g/L)^(1/2).
  • #1
Dekoy
14
0

Homework Statement


The speed of a rifle bullet may be measured by means of
a ballistic pendulum in the following way. The bullet, of
known mass m and unknown speed v, embeds itself in a
stationary wooden block of mass M, suspended as a pen-
dulum of length L. This sets the block to swinging. The
amplitude x of swing may be measured and, using conser-
vation of energy, the velocity of the block immediately after
impact may be found. Assume that x(double less than sign)L. Show that the
speed of the bullet is given by

Vb=[(m+M)/m]X(g/L)^(1/2)


Homework Equations


K=(1/2)mV^2
U=mgh
P=mV
Vw=velocity of wooden block

The Attempt at a Solution


Ok here's what I did I got part of the problem I went all the way to
Vb=[(M+m)/m](2gh)^(1/2)
The problem is going from (2gh)^(1/2) to X(g/L)^(1/2)
I used angular frequency and it seems to work but I'm not sure plus we haven't talked about that in class. Another way it seems to work is from radial acceleration but only if the radius = X which i think might be possible but not always. Watching a video from an MIT lecture I saw that it can be done using cosine expansion but I don't know how to.

Any help will be highly appreciate, thanks in advance.
 
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  • #2


Thank you for your post. It seems like you have made good progress in solving this problem. However, I would suggest taking a closer look at the conservation of energy equation that you have used.

The equation for conservation of energy in this situation would be:

K(initial) + U(initial) = K(final) + U(final)

Where K represents kinetic energy and U represents potential energy. In this case, the initial state is when the bullet is moving with an unknown speed v, and the final state is when the wooden block is swinging with a velocity Vw.

So, the equation would look like this:

(1/2)mv^2 + 0 = (1/2)(M+m)Vw^2 + mgh

Note that we have used the fact that the block starts at rest, so its initial kinetic energy is zero. Also, the potential energy of the block is mgh, where h is the initial height of the block.

Now, using the fact that the block swings to a maximum height of x, we can express the final potential energy as:

U(final) = (M+m)gx

Substituting this into our equation, we get:

(1/2)mv^2 = (1/2)(M+m)Vw^2 + (M+m)gx

Next, we can use the fact that the velocity of the block at the bottom of its swing is Vw = (2gx)^(1/2). This can be derived from the equation for simple harmonic motion, which I assume you have covered in class.

Substituting this into our equation, we get:

(1/2)mv^2 = (1/2)(M+m)(2gx) + (M+m)gx

Simplifying, we get:

(1/2)mv^2 = (M+m)(g+g)x

And finally, solving for v, we get:

v = [(M+m)/m]x(g/L)^(1/2)

Which is the same result that you have obtained. I hope this helps clarify any confusion you may have had. Keep up the good work!
 

1) What is a ballistic pendulum?

A ballistic pendulum is a device used to measure the speed of a bullet or other projectile. It consists of a pendulum with a known mass and length that is suspended from a pivot point. When a bullet is fired into the pendulum, it swings upward, and the height of the pendulum swing can be used to calculate the bullet's speed.

2) How does a ballistic pendulum measure the speed of a bullet?

The ballistic pendulum measures the speed of a bullet by using the principles of conservation of momentum and conservation of energy. The bullet's momentum and energy are transferred to the pendulum upon impact, causing it to swing upward. By measuring the height of the pendulum swing, the bullet's initial speed can be calculated.

3) What factors can affect the accuracy of a ballistic pendulum?

The accuracy of a ballistic pendulum can be affected by various factors, such as the mass and velocity of the bullet, the length and mass of the pendulum, and the pivot point's location. Air resistance and friction can also impact the pendulum's swing, so it is essential to minimize these factors for accurate results.

4) Can a ballistic pendulum be used to measure the speed of any type of projectile?

No, a ballistic pendulum is typically designed to measure the speed of small, high-velocity projectiles, such as bullets. It may not be accurate for measuring the speed of larger or slower-moving objects, as the pendulum's swing may not be significant enough to provide an accurate measurement.

5) What are the practical applications of measuring the speed of a bullet using a ballistic pendulum?

The speed of a bullet is crucial in forensic investigations, ballistics testing, and firearms development. By accurately measuring the bullet's speed, scientists and engineers can gather valuable data to improve bullet design and performance. It can also help in analyzing the trajectory and impact of a bullet in different scenarios, aiding in the investigation of crimes or accidents involving firearms.

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