## Impulse, Linear Momentum Newton’s Law Of Motion

In this article, we will discuss about impulse and linear momentum, Newton’s Law of Motion. As well as, Newton’s first law, second law and third law of motion. Moreover, with the influence of unbalanced forces upon bodies. The motion of bodies, and the laws governing such motions. Additionally, we will also study the collisions between bodies that are virtually free from forces other than those which they exert on each other at impact and the laws governing such collisions.

In addition, at the end of this chapter, we should be able to:

- State and explain on our own words the meaning of each of Newton’s three laws of motion.
- Solve simple problems involving the conservation of linear momentum.
- State and explain the meaning of the law of conservation of linear momentum.
- State and explain the meaning of the law of conservation of linear momentum.
- Verify those laws in the laboratory for an elastic collision of trolleys.
- Explain why (a) walking is possible (b) why a gun recoils when fired (c) how a rocket is propelled (d) how a jet plane is propelled.
- Solve simple problems based on Newton’s laws of motion and the principle of momentum.
- Explain why the weight of a body may vary from place to place.
- Explain inertial mass and the relationship between mass and weight.

## Impulse and Momentum

The momentum (p) of a body is defined as the product of its mass, m, and its velocity, v. Its unit is kgms-¹. Hence P = mv

Thus a body of mass 2 kg moving with a velocity of 10 ms- has the same momentum as a body of mass 5 kg moving with a velocity of 4ms-¹. Momentum is a vector. It has magnitude (the product m x v) and direction, the direction of v.

Momenta, like velocities, are therefore added by the parallelogram law. The product m x v is sometimes called linear momentum to distinguish it from a similar quantity called angular momentum.

Impulse is usually associated with the collision. When a body collides with another, each receives an impulse or blow. The impulse consists of a large varying force acting for a very short time. Additionally, we define Impulse as the product of the average force acting on a particle and the time during which it acts.

If a force F acts for a short time (t) the impulse (I) is given by 1 = F x t

The unit of impulse is the Newton-second. Impulse is a vector. It has the same direction as the direction of the force.

#### Example One

A stationary ball is hit by an average force 50 N for a time of 0.03 sec. What is the impulse experienced by the body?

#### Solution

I = Ft

= 50 × 0.03

= 1.5 N-sec

#### Example Two

A body of mass 3.0 kg moves with a velocity 10ms-¹. Calculate the momentum of the body.

#### Solution

p = mv

= 3 x 10

= 30.0 kgm^{–}s

## Newton’s Laws of Motion

In Book I Chapter 2, we stated that motion is caused by unbalanced forces. How forces are related to motion was first discovered by Sir Isaac Newton who also stated three important laws of motion known as Newton’s Laws of Motion.

## Newton’s First Law of Motion Idea of Inertia

The first law states that every object continues in its state of rest or of uniform motion in a straight line unless acted upon by an external force.

It is a common experience that a body at rest will remain at rest. For example, a book at rest on top of a reading table will remain there unless something pushes it or pulls at it. And also, as was first pointed out by Galileo, the motion of a body once started, would continue along a straight line path unless some forces cause it to change. This means that a moving object, if left to itself, will move in the same direction forever. That is, once a body has been set in motion it is no longer necessary to exert a force on it to maintain it in motion. But our everyday experience does not seem to agree with this. We know that any sliding object slows to a stop. Why? Because friction or gravity or other external forces slow moving objects to a stop. But in the absence of any unbalanced force, an object would move forever with the same velocity. It would neither slow down nor deflect.

This tendency of bodies to remain in their state of rest or of uniform linear motion in the absence of applied forces is known as Inertia.

Newton’s first law shows that inertia is inherent in a body at rest or the one moving with a constant velocity.

Inertia is a property of matter. Mass is a measure of inertia, the more massive an object is, the more inertia it has.

Newton’s first law explains why passengers in a fast moving vehicle tend to move forward when the car suddenly stops, or to be pushed backwards when the vehicle suddenly speeds off, since there is little or no force to restrain them. Hence safety belts are used to provide this restraining force.

Newton’s first law also explains what force does, but does not suggest how force should be measured.

## Newton’s Second Law of Motion

The Newton’s second law of motion states that the rate of change of momentum is proportional to the impressed force and takes place in the direction of that force.

Newton’s second law enables us to define absolute unit of force which remains constant under all conditions. In symbols the law states that

Fx = Change in momentum/Time taken for the change

F = mv-mu/t

Where m, u, v, t are the mass, initial velocity, final Velocity and time respectively of motion of the body acted upon by a force F; and the product of m and v is called the momentum.

In the SI system, F is in Newton, m is in kilogram and the acceleration, a, is in meter per square second (ms-2). The Newton is the unit of force which gives a mass of 1 kg an acceleration of 1 ms^{–}2

Thus,

F= m (v – u)/t

The product F-t is the impulse, I of the force. 1 = Ft = change in momentum.

Unit of I is Newton-second (Ns). This is also a unit of momentum.

We see that Newton’s second law of motion gives a measure of force as the product of mass and the acceleration of a body. Hence acceleration, a, is given by

This second law of motion also gives an operational definition of Force as the rate of change of momentum with time.

#### Example One

An unbalanced force of 20 N acts on a 4.0 kg mass. What acceleration does it give it?

#### Solution

F = ma

20N = 4a

a = 5ms^{–}2

#### Example Two

A 900g stone is pushed along a tarmac by a horizontal force of 20 N. A frictional force of 8 N opposes the motion. What is the acceleration given to the stone?

#### Solution

Unbalanced force = (20-8) N = 12 N

F = ma

12 N = 900/1000kg x a=0.9a

a= 12/0.9 = 13.33 ms-2

## Newton’s Third Law of Motion

The third law of motion states that Action and Reaction are equal and opposite. Or To every Action there is an equal and opposite Reaction.

The law implies that when a body A exerts a force F_{A} on a body B, the body B always exerts a force F_{B} on the body A. The forces F_{A} and F_{B} are equal in magnitude but opposite in direction and since force is a vector, we can write F_{A} = – F_{B}. F_{A} is the action force, F_{B} is the reaction force. For example, when you push down on your desk with your fingers, the desk pushes up on your fingers with an equal force. If you hit a wall with your head, the force exerted by your head acts on the wall; at the same time the wall exerts an equal and opposite force on your head. You will feel this reaction as pain in your head. In the above examples, either force may be considered the action or the other treated as the reaction.