1. Mathematically explain in Detail how Newton's cradles work. Be able to use your understanding of...

1. Mathematically explain in Detail how Newton's cradles work. Be able to use your understanding of how Newton's cradles work to create a physics equation/formula that can be used in any situation.

Expert Solution

Newton's cradle works based on conservation of energy and momentum.
Consider the simplest Newton's cradle with only two spheres. Consider that each sphere has a mass m.
The first sphere is raised and released. It will have a velocity V. When the first sphere hits the second one, consider that the second sphere move with velocity Vs and the first sphere recoil with a velocity Vr
Using conservation of momentum, mV = mVs - mVr
V = Vs - Vr ...(1)

Using conservation of energy,
1/2 mV² = 1/2 mVs² + 1/2 mVr²
V² = Vs² + Vr² ...(2)

Substituting (1) in (2),
(Vs - Vr)2 = Vs² + Vr² ..
Vs² + Vr² - 2Vs Vr = Vs² + Vr²
- 2VsVr = 0
Vr = 0 since Vs can not be zero.
So, there is no recoil velocity for the first mass. This can be extended to Newton's cradle with many spheres. When the momentum and energy is conserved, velocity will be exchanged. The sphere at rest exchange its velocity with the sphere that hits it.

genius_generous answered 2 months ago

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