# Solved: Ballistic Pendulum Velocity Calculation

• Arejang
In summary: I just wanted to see if I were on the right track or not. Thanks again for all your help!In summary, the problem involved a bullet of mass 0.01 kg colliding with a block of wood of mass 1.5 kg suspended as a pendulum. After the collision, the block and bullet swing upward a distance of 0.40 m. The goal was to find the initial velocity of the bullet. The solution involved using both momentum and energy conservation equations, with the final answer being 423 m/s. However, the given answer choices were all incorrect, possibly due to the dissipation of energy in the collision. Further clarification from the instructor is recommended.
Arejang
[SOLVED] A Ballistic Pendulum

## Homework Statement

A bullet of mass 0.01 kg moving horizontally strikes a block of wood of mass 1.5 kg which is suspended as a pendulum. The bullet lodges in the wood, and together they swing upward a distance of 0.40 m. What was the velocity of the bullet just before it struck the wooden block? The length of the string is 2 meters.

## Homework Equations

Momentum Conservation:

$$m_{a}v_{a1}+m_{b}v_{b1}=(m_{a}+m_{b})*v_{2}$$

Energy Conservation:

$$1/2mv^{2}=mgy$$

## The Attempt at a Solution

Since the block is at rest before the bullet hit, if we use the momentum conservation formula, we only have to deal with the initial speed of the bullet. The resultant formula is

$$m_{b}v_{b}=(m_{w}+m_{b})*v_{2}$$

Once the bullet is embedded in the block, it will have a potential energy of zero and a kinetic energy of

$$K=1/2(m_{b}+m_{w})v_{2}^{2}$$

and the block with the bullet travels up a height .40m, and comes to a rest. At this point, the block/bullet unit has a kinetic energy of zero, and a potential energy of

$$U=(m_{b}+m_{w})gy$$

Using energy conservation we get:

$$1/2(m_{b}+m_{w})v_{2}^{2}=(m_{b}+m_{w})gy$$

we can solve for velocity here and get the speed after the bullet hit the block. The masses should cancel out, leaving:

$$v_{2}=\sqrt{2gy}$$

We sub in this expression for v back into the first momentum formula, getting:

$$m_{b}v_{b}=(m_{b}+m_{w})*\sqrt{2gy}$$

solving for the initial velocity $$v_{b}$$, we get $$v_{b}=(m_{b}+m_{w})/m_{b}*\sqrt{2gy}$$

At this point I just plugged and chugged, using the given values in the problem and came up with 423 m/s, but it turned out to be the wrong answer. Can anyone help me figure out what I did wrong? Much thanks in advance!

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If the bullet lodged into the wood, you can't use conservation of Kinetic Energy. Rather than that...try using what you know about centripetal forces and acceleration.

How could you use centripetal force if the pendulum isn't moving in a circle?

Arejang said:
At this point I just plugged and chugged, using the given values in the problem and came up with 423 m/s, but it turned out to be the wrong answer. Can anyone help me figure out what I did wrong? Much thanks in advance!
Your solution looks good to me. Who says it's wrong?

Our assignments are given online and it grades it the instant you input an answer. In this case, the answers are given in multiple choice, so I'm usually cautious to choose an answer as the chances to get it right are obviously limited to the amount of choices available. But I did choose the 423 m/s and it was shown as incorrect.

Here are my answer choices:

250 m/s
423 m/s
66.7 m/s
646 m/s
366 m/s

And yes, I felt the same way too, doc. I'm pretty sure I didn't fumble on any of the calculations and that I used the correct formula to derive the speed, unless "the initial speed of the bullet" and "the speed of the bullet just before it hits the block" mean totally different things.

Wait...I see what you did there...nevermind...everything looks to be alright...dont know what's wrong with your answer.

Last edited:
Gear300 said:
For inelastic collisions, energy is lost in the collision due to deformation. Because of that, Ki = Pf + Ediss...To avoid dealing with the dissipated energy, the best thing to do would be to use the conservation of momentum. What you do is set Pf = Ka, in which Ka is the energy right after collision (because gravitational energy is interconvertable with kinetic energy). So it would be (1/2)*(Ma + Mb)*(v2)^2 = (Ma+Mb)gh. Solve for v2, plug it into your momentum equation, and then solve for v1.
That's exactly what Arejang did!

Arejang said:
I'm pretty sure I didn't fumble on any of the calculations and that I used the correct formula to derive the speed, unless "the initial speed of the bullet" and "the speed of the bullet just before it hits the block" mean totally different things.
I doubled checked your arithmetic; I would have chosen the same answer.

I just guessed all the answers and the right one was 66.7 m/s...What the heck?

Sorry, this problem has been irking me to no end. Would the length of the string, 2m, play any part of this problem? That would be the only factor I could see that might alter the answer somewhat. But if that were the case, then I don't know how else to approach this problem.

Arejang said:
Would the length of the string, 2m, play any part of this problem?
Not that I can see. Your solution is perfectly correct.

thanks, I'm going to go ahead and marked this solved for the time being. If I find anything else, I'll post it back up again. Thanks for all your help!

If this homework is graded (or even if it's not), make sure your instructor sees your solution.

I plan on doing so when I see him tomorrow.

## What is a ballistic pendulum velocity calculation?

A ballistic pendulum velocity calculation is a method used to determine the initial velocity of a projectile by measuring the height to which a pendulum is lifted after the projectile collides with it.

## How does a ballistic pendulum work?

A ballistic pendulum consists of a pendulum bob suspended from a rigid frame. When a projectile hits the bob, it transfers its momentum to the pendulum, causing it to swing upward. By measuring the height of the pendulum after the collision, the initial velocity of the projectile can be calculated.

## What factors can affect the accuracy of a ballistic pendulum velocity calculation?

The accuracy of a ballistic pendulum velocity calculation can be affected by factors such as air resistance, friction in the pendulum's pivot, and the weight and shape of the pendulum bob. These factors can cause the pendulum to not swing as high as predicted and result in a lower calculated velocity.

## What are some applications of ballistic pendulum velocity calculations?

Ballistic pendulum velocity calculations have various applications, including determining the muzzle velocity of firearms, measuring the speed of projectiles in ballistics and forensics, and studying the conservation of momentum in physics experiments.

## Are there any limitations to using a ballistic pendulum for velocity calculations?

Yes, there are some limitations to using a ballistic pendulum for velocity calculations. For example, it is only accurate for objects with a known mass, and it is not suitable for very high-velocity projectiles as they can cause damage to the pendulum or exceed its measuring range. Additionally, it assumes a perfectly elastic collision between the projectile and the pendulum, which may not always be the case in real-world situations.

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