# Bullet hitting a block on a string. Length of the string no effect?

#### johnwandering

##### Full Member
10+ Year Member
In the classic illustration of a bullet hiting a block suspended by a string, BR notes that the length of the string makes no difference on the change in height the bullet climbs.

How does this make sense?

If a string is only 3cm long, the block can only climb 6cm at the most (assuming circular path).
But if the string is a 5cm long, the same bullet can cause the block to rise maybe 10cm.

So... BR is wrong? Right?
I dont see any 2 ways around this.
Change in the L string length obviously seems to change height bullet climbs.

#### MrNeuro

##### Full Member
7+ Year Member
In the classic illustration of a bullet hiting a block suspended by a string, BR notes that the length of the string makes no difference on the change in height the bullet climbs.

How does this make sense?

If a string is only 3cm long, the block can only climb 6cm at the most (assuming circular path).
But if the string is a 5cm long, the same bullet can cause the block to rise maybe 10cm.

So... BR is wrong? Right?
I dont see any 2 ways around this.
Change in the L string length obviously seems to change height bullet climbs.
i posted the same question

easiest way is elastic collisions mean total energy is conserved

1/2 mv^2 = mgh
h = v^2/ 2*g

height isn't affected by length

2nd way

mgL(1-costheta) is the potential energy

w/ a longer string theta is going to be smaller cos theta will be larger
so
Large Length (1-large number) = large (small)
shorter string theta will have to be larger cos theta will be smaller
so
small length(1-small number) = small(large)

#### milski

##### 1K member
5+ Year Member
Ok, you're taking it a bit too literally. The block will climb the same height, provided that the string is low enough to allow it to climb that high. It's momentum conservation + energy conservation problem. The momentum of the bullet mv, is transfered to the block and gives it some velocity, V=mv/M. That corresponds to some amount of KE, MV^2/2 which gets transformed in potential energy. Since the potential energy is U=Mgh=MV^2/2 or h=V^2/(2g). As you see, the height does not depend on the length of the string. All these assumes that the string is long enough and allows the block to raise high enough to convert all the KE to PE. If the string is too short for that, the block will end up rotating around the other end of the string.

#### MrNeuro

##### Full Member
7+ Year Member
Ok, you're taking it a bit too literally. The block will climb the same height, provided that the string is low enough to allow it to climb that high. It's momentum conservation + energy conservation problem. The momentum of the bullet mv, is transfered to the block and gives it some velocity, V=mv/M. That corresponds to some amount of KE, MV^2/2 which gets transformed in potential energy. Since the potential energy is U=Mgh=MV^2/2 or h=V^2/(2g). As you see, the height does not depend on the length of the string. All these assumes that the string is long enough and allows the block to raise high enough to convert all the KE to PE. If the string is too short for that, the block will end up rotating around the other end of the string.
question milski

how can you apply theta = S/R to this problem or is that not possible?

#### milski

##### 1K member
5+ Year Member
question milski

how can you apply theta = S/R to this problem or is that not possible?

What is &#952;=S/R? Does not ring a bell on its own.

If you want to deal with the motion of the block from kinematics point of view, it will be extremely painful, especially since you cannot estimations like sin&#952;=&#952;. Or do you have something completely different in mind?

#### MrNeuro

##### Full Member
7+ Year Member
What is &#952;=S/R? Does not ring a bell on its own.

If you want to deal with the motion of the block from kinematics point of view, it will be extremely painful, especially since you cannot estimations like sin&#952;=&#952;. Or do you have something completely different in mind?

theta is angle subteneded by the radius
s is arclength travelled by theta
theta is angle travelled by radius

i think you can use it b/c the pendulum formulas assume that theta is a small angle and they assume sin theta = tan theta

also think this may help

http://interactagram.com/physics/kinamatics/pendulum/

Last edited:

#### milski

##### 1K member
5+ Year Member
theta is angle subteneded by the radius
s is arclength travelled by theta
theta is angle travelled by radius

i think you can use it b/c the pendulum formulas assume that theta is a small angle and they assume sin theta = tan theta

Ah, ok. &#952;=s/r is not an estimation - it's always correct. But how are you going to apply it to this problem? You know &#952; and r, that will allow you to find s, but then the motion along s is not easy to describe, so you cannot calculate anything from there.

The pendulum formulas from intro physics do assume small angle and &#952;=sin&#952;=tan&#952;. By small &#952; you should think of 10 degrees or even less, depending on how much error you are willing to take. You certainly cannot extrapolate that for larger angles.

The constant height of how much the block raises on the other hand is exact - it does not change (for long enough R).

#### MrNeuro

##### Full Member
7+ Year Member
Ah, ok. &#952;=s/r is not an estimation - it's always correct. But how are you going to apply it to this problem? You know &#952; and r, that will allow you to find s, but then the motion along s is not easy to describe, so you cannot calculate anything from there.

The pendulum formulas from intro physics do assume small angle and &#952;=sin&#952;=tan&#952;. By small &#952; you should think of 10 degrees or even less, depending on how much error you are willing to take. You certainly cannot extrapolate that for larger angles.

The constant height of how much the block raises on the other hand is exact - it does not change (for long enough R).

thanks

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