# Mathematical Formulation of Second Law of Motion

**Newtonâ€™s Second Law of Motion**

A football bouncing off the ground and hitting a glass window slowly will not break it. But if someone gives it a hard kick and it travels fast and hits the same window, the glass will break. (This is only one example, not an experiment so you don't have to actually see what happens). Though the football has the same mass both times, it has a greater quantity of motion in the second time because it is traveling with a greater velocity.

In an accident a huge truck will cause more damage than a car even if both were moving with the same velocity before the crash. We can explain it by saying that though the truck and car were traveling at the same velocity, the truck had a greater quantity of motion because it had more mass than the car.

A small mass such as a bullet can kill a person when fired from a gun, because of its great velocity. Similarly, a fast moving table tennis ball cannot injure a person when it hits him but a fast moving cricket ball can cause severe injury to a player or spectator.

We see that the quantity of motion a body possesses depends on both its mass and velocity. In Physics, we use the word momentum for the quantity of motion a body has.

Momentum of a body is the product of its mass and its velocity.

Momentum p = mass x velocity = mv

(kg m/s is the unit of momentum)

Momentum is a vector quantity and has the same direction as that of velocity.

Newton's second law of motion explains the concept of force

Force is defined as that which when acting on a body changes or tends to change the state of rest or of uniform motion of the body.

Momentum of a body is the product of its mass and velocity. Obviously, it is the measure of quantity of motion of a body. The unit of momentum is kg m/s.

Newton's second law of motion states that, "the rate of change of the momentum of a body is proportional to the impressed force acting on it and it takes place in the direction of force".

Newton's second law gives the relation between force, mass and acceleration.

Consider a body of mass 'm' moving with initial velocity 'u' and attaining the final velocity 'v' in a time 't' seconds under the applications of a force.

Initial momentum of the body = mu

Final momentum of the body = mv

Change in momentum = mv â€“ mu

Rate of change of momentum =

= m

= ma [Q a = ]

According to Newton's second law, the rate of change of momentum is directly proportional to force.

F ∝ ma or ** **F = k ma

where k is a constant of proportionality. Its value depends on the choice of the unit in which the force is to be measured.

When m = 1, a = 1 and F = 1

**Example:**

1 = k × 1 × 1

1 = k

∴ F = ma

Thus the second law of motion gives us a method of measuring the force acting on an object as a product of its mass and acceleration.

If m = 1 kg and a = 1 m/s^{2} then F= 1N.

We can now define 1N as that force which when acting on a body of mass 1 kg produces an acceleration of 1 m/s^{2} in it.