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04 Colliding Balls (1)

Aim

To demonstrate many combinations of elastic collisions. To test momentum conservation.

Subjects

Diagram

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Figure 1:.

Equipment

Elastic balls hanging from a frame (“Newton’s cradle”)

Presentation

The identical balls are bifilarly suspended in a straight row. In horizontal equilibrium the balls are just in contact. Speeds at the time of contact are, to a first approximation, proportional to the horizontal displacement from rest position.

  1. Two balls are suspended. One ball is pulled out and released. It hits the other and this one bounces out to the other side; the first ball being at rest now. (See Figure 2.)

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Figure 2:.

Both balls are pulled out and released. They hit and both rebounce almost to the original height. (See Figure 3.)

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Figure 3:.

Both balls are pulled out but one ball more than the other. In this way the two balls will have different speeds. They are released, hit, and it can be observed that after the collision the two balls have interchanged their speeds. (See Figure 4.)

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Figure 4:.

  1. More balls are suspended.

When 3 (or 4 , or 5 , etc.) balls are suspended the demonstrations performed with two balls can be repeated. The observed phenomena are similar. The balls between the two outer balls are not taking part in the movements. (See Figure 5.)

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Figure 5:.

When two balls are pulled out and released, they hit the others and two balls bounce out to the other side. With three, three bounce out, etc. It is always the same number of balls. (See Figure 6.)

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Figure 6:.

  1. Two balls of different mass ( mm and 3m3 m ).

    1. mm is hanging in its rest position and 3m3 m is pulled out and released. After the collision mm is launched and 3m3 m follows it but with reduced speed. After reversing their direction, they collide again in the middle and after this second collision mm hangs at rest and 3m3 m rebounces to its original height.

    2. 3m3 m is hanging in its rest position and mm is pulled out and released. After the collision, 3m3 m is launched and mm rebounces but not to its starting position. Both masses have the same speed when they collide again exactly in the rest position and after this second collision 3m3 m is at rest and mm rebounces to its starting position.

    3. Both balls are pulled out and released at the same height, so they have the same speed. After the collision, 3m3 m is at rest and mm bounces to a much higher height. mm reverses its direction and there is a second collision. Now both balls rebounce to their original height.

In all three demonstrations the starting position returns after two collisions.

Explanation

Figure 7 shows the situation before and after an elastic collision. In an elastic collision, both momentum and kinetic energy are conserved. This means that vrel v_{\text {rel }} will not change: u2−u1=Vrel u_{2}-u_{1}=V_{\text {rel }}

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Figure 7:.

Momentum is conserved:

m1u1+m2u2=m1vrel m_{1} u_{1}+m_{2} u_{2}=m_{1} v_{\text {rel }}
(1)

These two equations give us:

u1=m1−m2m1+m2vrel u_{1}=\frac{m_{1}-m_{2}}{m_{1}+m_{2}} v_{\text {rel }}
(2)

and

u2=2m1m1+m2vrel u_{2}=\frac{2 m_{1}}{m_{1}+m_{2}} v_{\text {rel }}
(3)

The next table shows the results of different situations concerning our demonstrations.

m1/m2m_{1} / m_{2}u1\mathrm{u}_{1}u2\mathrm{u}_{2}
1/31 / 3−1/2v-1 / 2 v1/2v‾\underline{1 / 2 v}
10v‾\underline{v}
31/2v1 / 2 \quad v3/2v‾\underline{3 / 2 v}

This table explains the behavior shown in presentation 1 and 3 .

In the demonstration of Figure 5, aa hits bb. bb gets the speed of aa (see table) and immediately hits cc. bb comes to rest and cc gets the speed of bb and immediately hits dd etc. At the end, ee is launched with the speed vv that aa originally had.

In the demonstration of Figure 6 , the first thing that happens is that bb hits cc. bb comes to rest and finally ee is launched. In the meantime aa hits bb and aa comes to rest and finally dd is launched.

If not reasoning step by step, the question could be raised why ball ee is not coming out with double velocity. After all this would conserve momentum. But checking kinetic energy will show that in that case kinetic energy is not conserved.

Remarks

In succession to demonstration 2 (in the end with three balls), one ball with mass 3 m3 \mathrm{~m} and speed vv hits the row of balls at rest. It can be seen that in this case not the last three balls are launched with speed vv, but that all balls are moving now with different speeds and also the 3m3 m-ball is still moving.

Sources

1N20 Conservation of Momentum
03 Explosion
1N20 Conservation of Momentum
05 Demonstrator and Cart