# Relation Between Kinetic Energy and Momentum

## Kinetic Energy

The kinetic energy of an object is the energy associated with the object which is under motion. It is defined as “the energy required by a body to accelerate from rest to stated velocity.” It is a vector quantity.

### Kinetic Energy formula

Mathematically expressed as-

$$K.E=\frac{1}{2}mv^{2}$$

Where,

m is the mass of the object measured in kg.

v is the velocity of the object measured in m/s

### Kinetic Energy unit

The SI Unit of Kinetic Energy is Joules.

## Momentum

The momentum of an object is the virtue of its mass. It is defined as the product of mass and velocity. It is a vector quantity.

### Momentum formula

The momentum of a moving object can be mathematically expressed as –

$$p=mv$$

Where,

• p is the momentum.
• m is the mass of the object measure using kg.
• v is the velocity of object measure using m/s.

### Momentum unit

The SI Unit of Momentum is kg.m/s

## Relation Between Momentum and Kinetic Energy

Kinetic energy and momentum of a moving body can be mathematically related as follows-

Consider the formula of kinetic energy-

$$K.E=\frac{1}{2}mv^{2}$$

Multiply and divide R.H.S by m

$$\Rightarrow K.E=\frac{1}{2}mv^{2}\times \frac{m}{m}$$ $$=\frac{m^{2}v^{2}}{2m}$$ $$=\frac{\left ( mv \right )^{2}}{2m}$$

We know that $$p=mv$$. Substituting for mv in the above equation we get-

 $$K.E=\frac{p^{2}}{2m}$$

The above equation gives the relation between Kinetic energy and momentum of the object which is under motion.

 If the Momentum(p) is constant $$K.E\propto \frac{1}{m}$$ If Kinetic Energy(K.E) is constant $$p\propto \sqrt{m}$$ If mass(m) is constant $$p\propto \sqrt{K.E}$$

Hope you have understood the relation between Kinetic energy and momentum of a moving object.

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