# Ballistic Pendulum and Projectile Motion

#### jkb

1. Homework Statement

The question involves a bullet weighing 0.04 kg and traveling at 360 m/s being fired at a pendulum ball (weighing 1.0 kg) with which it becomes lodged and the string (which is 2 m long) thereafter breaks at 45 degrees to the ground. The ball then acts with projectile motion traveling initially at 45 degrees and lands some distance away. What is the total horizontal distance traveled?

2. Homework Equations

KE= .5*m*v^2
PE= m*g*h
*conservation of energy*

Kinematic Equations

3. The Attempt at a Solution

I would be able to find the range that the ball travels after the string breaks away but I am unable to find a value for the velocity of the ball at this point as well as at what height the ball would be at 45 degrees. I am thinking that it would just be 1 m (since at 90 degrees it would be 2 m above the ground) but I am very confused about where to go. I know that using the conservation of mass I can use the initial speed of the bullet and the weight of the bullet along with the weight of the ball [ m1v1 = m2v2 ] to find the velocity to be 13.85 m/s but how does that velocity change at 45 degrees, how far does the ball travel horizontally in that period and at what height above the ground is the ball when the string breaks (at 45 degrees)? After this could be cleared up for me a bit I would be able to attempt this problem much easier. I am tutoring a fellow student in this class tomorrow and this problem is due when after the weekend so your help would be much appreciated 1. Homework Statement

2. Homework Equations

3. The Attempt at a Solution

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#### cristo

Staff Emeritus
From the way the question is worded, it seems that when the string is 45 deg to the ground, it breaks, and the mass starts a projectile motion at 45 deg to the ground.

I'm not sure what vertical distance you are looking for, however, since in the question you do not state where the ground is. My best *guess* is that the pendulum is just resting on the ground when it is hit with the bullet, and so then the height of it when the string breaks is 2m-vertical distance when string breaks. This last distance can be calculated using trigonometry (draw a diagram!)

Then to find the intial velocity of the projectile, you can use conservation of momentum.

#### jkb

It is asking the total distance traveled by the mass therefore the horizontal distance it travels while moving in the pendulum motion as well as the range it travels after the string breaks.

The ground is where the ball is initially resting on and the vertical distance I am looking for it the vertical distance the ball is at exactly when the string breaks and it starts the projectile motion.

Also, I am looking for the velocity at this point because the velocity I found is very small and makes little sense.

I found out how I could find the horizontal distance traveled during the pendulum motion using trigonometry but the new question I have is that if the ball is at a 45 degree angle when it breaks, is it simply at 1 vertical meter above the ground? I thought this because since the string is 2 m long and if the ball was at 90 degrees and it broke it would be 2 m off the ground but since it breaks at 45 degrees it would be 1 m off the ground??

I am looking at this question in so many different ways that I am probably confusing you as well lol

Using the fact that the ball is 1 meter off the ground at 45 degrees (if that is true) then I found a time using the D = 1 m and Vo= 13.85 (i found using momentum conservation) and A = -9.8 to find a time to travel that distance and then using that time I used Vf= Vo + At to find the velocity when the string breaks but the answer seems very unreasonable.

Thanks for any help u can provide!!

#### cristo

Staff Emeritus
It is asking the total distance traveled by the mass therefore the horizontal distance it travels while moving in the pendulum motion as well as the range it travels after the string breaks.
Id say yes, it wants the total distance travelled.

The ground is where the ball is initially resting on and the vertical distance I am looking for it the vertical distance the ball is at exactly when the string breaks and it starts the projectile motion.

Also, I am looking for the velocity at this point because the velocity I found is very small and makes little sense.

I found out how I could find the horizontal distance traveled during the pendulum motion using trigonometry but the new question I have is that if the ball is at a 45 degree angle when it breaks, is it simply at 1 vertical meter above the ground? I thought this because since the string is 2 m long and if the ball was at 90 degrees and it broke it would be 2 m off the ground but since it breaks at 45 degrees it would be 1 m off the ground??
No, the vertical distance will be 2-(2sin45). You need to use trigonometry. (Note that, since the angle is 45, the vertical distance from the pivot to the end of the bob when the string snaps is the same as the horizontal distance when the string snaps.

Using the fact that the ball is 1 meter off the ground at 45 degrees (if that is true) then I found a time using the D = 1 m and Vo= 13.85 (i found using momentum conservation) and A = -9.8 to find a time to travel that distance and then using that time I used Vf= Vo + At to find the velocity when the string breaks but the answer seems very unreasonable.
You have used the equations for projectile motion, when the pendulum is not acting as a projectile! You used the momentum conservation to calculate the velocity. This is the velocity when the string snaps. Use this in the projectile motion equations as the initial velocity.