# Newton’s Second Law of Motion: Definition, Derivation, and Application

You have already studied Newton’s First Law of Motion, In this blog, we will discuss Newton’s Second Law of Motion, derive the second law of motion, and its applications.

According to Newton’s Second Law of Motion: The rate of change of linear momentum of a body is directly proportional to the external force applied on the body and this always takes place in the direction of the force applied.

What is momentum?

From the first law of motion, we know that when an unbalanced external force acts on an object, its velocity changes and the object accelerates.

How this acceleration is related to the force applied?

We know that a truck at rest does not require any attention when parked along the roadside, but a moving truck even at a speed as low as 5m/s may kill a person coming in its path.

A bullet of small mass fired from a gun may go deep into the wood because of its large velocity

This suggests that the impact produced by objects depends on their
• Mass and
• Velocity

Newton introduced a physical quantity which is a combination of both mass and velocity of an object. This quantity is known as Momentum

The Linear momentum (p) of an object is defined as the product of its mass and velocity
p = mv

The unit of momentum is kg * m/s
If a body is at rest or we can say that its velocity is zero then its momentum will also be zero

Linear momentum is a vector quantity that is the product of both magnitude and direction. The direction of linear momentum is the same as that of velocity. As the first law of motion suggests that a force is required to change the velocity of a body, so when the velocity of a body changes, its linear momentum also changes.

We can express Newton’s Second Law of Motion as:- The rate of change in linear momentum/time taken is proportional to the force applied. This means (p2- p1)/t is proportional to F. It means that when a bigger force is applied to a body its linear momentum changes at a faster rate which means it takes lesser time for the momentum to change if the applied force is larger.

Let’s derive newton’s second law of motion.

Mathematical formulation of Newton’s Second Law of Motion

Suppose,
m is the mass of a body
u is the initial velocity of the body along a straight line
F is an external force applied on the body, which is constant in magnitude
t is the time for which the force is applied
v is the final velocity of the body along the same straight line, after ‘t’ seconds

Initial linear momentum of the body p1 = mu
Final linear momentum of the body = p2 = mv
Change in linear momentum of the body = p2 –p1
= mv – mu
= m (v-u)

Rate of change of linear momentum
= change in linear momentum/time taken
= m (v-u)/t

Also we know that rate of change of velocity is acceleration i.e. v-u/t = a
Therefore Rate of change of linear momentum = ma

According to Newton’s second law of motion:- The rate of change of linear momentum is proportional to the force applied or ma is proportional to F.

This implies F = Kma
Where, K is the constant of proportionately

Unit force is that much force which when applied on a body of unit mass produces unit acceleration
That is if m=1, a=1, F=1

Therefore 1= K * 1 * 1 this implies, K=1

So putting K=1 in the above equation we have, F = ma

This is the mathematical form of Newton’s second law of motion. It states that- Force acting on a body is the product of mass of the body and the acceleration of the body

Unit of force = Newton and symbolized using ‘N’. So, one Newton force is that much force which when applied on a body of mass 1 kg produces an acceleration of 1 meter per second square. Force is a vector quantity. Its direction is the same as that of the acceleration of the body

Application of Newton’s Second Law of Motion

The applications of the second law of motion are often seen in the actions in our everyday life.

1. To catch a fast cricket ball a player pulls his hands backward to prevent injury to his hands. By doing so the player increases the time during which the high velocity of the cricket ball decreases and the player has to apply a smaller force against the ball in order to stop it. The ball in turn exerts a smaller force on his hands and the hands are not injured.

If the ball is stopped suddenly, the high velocity of the ball would be reduced to zero in a very short interval of time. Therefore, the rate of change of linear momentum of the ball would be large and therefore a large force would have to be applied to catch the ball which might hurt the hands of the player.

2. Now take the case of the high jump. In the event of a high jump, the athletes are made to fall either on a cushioned bed or on a sand bed. Falling on a cushioned bed or on a sand bed increases the time during which the velocity of the athletes would be reduced to zero.

This decreases the rate of change of momentum of the athlete and hence the force on the athlete. The injury to the athlete is thus avoided.