Physics

To determine the ratio of the momenta when an object falls from rest and air resistance is ignored, we can use the relationship between momentum, velocity, and the motion of the object.

 

Velocity of the object as a function of distance

The object is dropped from rest, so its initial velocity . As the object falls under the influence of gravity, its velocity increases due to acceleration. The velocity at a given height is given by the kinematic equation for free fall:

where is the acceleration due to gravity, and is the height from which the object has fallen.

 

Momentum of the object

The momentum of an object is given by:

where is the mass of the object and is its velocity.

 

For a fall of height , the velocity is , so the momentum is:

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