Question

A light object and a heavy object have the same kinetic energy. Which has the greater momentum? Explain.

Short answer

The heavier object has greater momentum than the lighter object.

Step-by-step solution

Momentum

The product of mass and velocity of an object is termed the momentum of that object. It is denoted by the symbol p. It is a vector quantity.

For example, if an object of mass m is moving with the velocity v, its momentum is given as \(p = mv\).

Determining the relation between momentum, mass, and kinetic energy

If an object of mass m moves with the velocity v, its kinetic energy is

\(KE = \frac{1}{2}m{v^2}\).

On multiplying the numerator and denominator of the right-hand side of the above equation by m,

\(\begin{aligned}{c}KE = \frac{1}{2}m{v^2} \times \frac{m}{m}\\ = \frac{1}{{2m}}{m^2}{v^2}\\ = \frac{1}{{2m}}{\left( {mv} \right)^2}\end{aligned}\)

The momentum of the object is\(p = mv\).

Thus, the equation of kinetic energy can be written as follows:

\(\begin{aligned}{l}KE = \frac{1}{{2m}}{p^2}\\{p^2} = 2m\left( {KE} \right)\\p = \sqrt {2m\left( {KE} \right)} \end{aligned}\)

This is the relation between momentum, kinetic energy, and mass

Determining an object with greater momentum

It is clear from the above equation that the momentum of an object is directly proportional to the square root of its kinetic energy and the square root of its mass.

Since the kinetic energy of the lighter and heavier object is the same. Thus, the momentum of the object will depend on its mass. So, the object with greater mass will have greater momentum.

Therefore, the heavier object with greater mass has greater momentum than the lighter object with a smaller mass.