Newton’s cradle is a toy that consists of several metal balls touching each other and sus-pended on strings. When you pull one ball to the side and let it strike the next ball, only one ball swings out on the other side. When you use two balls to hit the others, two balls swing out. Can you account for this effect using your knowledge about elas-tic collisions?
The Correct Answer and Explanation is :
Yes! The effect observed in Newton’s Cradle can be explained using the principles of elastic collisions and the conservation of momentum and kinetic energy.
Explanation:
1. Conservation of Momentum:
Momentum (( p )) is given by:
[
p = mv
]
where ( m ) is the mass of the ball and ( v ) is its velocity. In an isolated system (like Newton’s Cradle), the total momentum before and after the collision remains constant.
If one ball is pulled back and released, it strikes the stationary balls. Let’s assume the mass of each ball is m and the velocity of the first ball before impact is v. The initial momentum is:
[
p_{\text{initial}} = mv
]
After the collision, if one ball were to move with velocity ( v ), the total final momentum would be:
[
p_{\text{final}} = mv
]
which satisfies the conservation of momentum.
If two balls are pulled back, their combined initial momentum is:
[
p_{\text{initial}} = 2mv
]
For momentum to be conserved, two balls on the other side must move with the same velocity ( v ), keeping the total momentum ( 2mv ).
2. Conservation of Kinetic Energy:
Since the collisions in Newton’s Cradle are elastic, kinetic energy (( KE )) is also conserved:
[
KE = \frac{1}{2} mv^2
]
For one ball, the total initial kinetic energy is:
[
KE_{\text{initial}} = \frac{1}{2} mv^2
]
The final kinetic energy, when one ball moves at ( v ), is:
[
KE_{\text{final}} = \frac{1}{2} mv^2
]
For two balls moving at velocity ( v ), their final kinetic energy is:
[
KE_{\text{final}} = 2 \times \frac{1}{2} mv^2 = mv^2
]
Thus, energy is also conserved.
Conclusion:
- The system follows the laws of elastic collisions.
- Both momentum and kinetic energy are conserved.
- This is why the same number of balls swing out on the other side as were initially pulled back.