Newton’s Law of Motion – SS2 Physics Lesson Note (First Term)

Newton’s laws of motion describe how forces change the motion of bodies. The first law says a body stays at rest or in uniform motion in a straight line unless a force acts on it. The second law says force equals mass times acceleration. The third law says every action has an equal and opposite reaction.

Behavioural objectives

By the end of the lesson, students should be able to:

1. State Newton’s three laws of motion.
2. Define inertia, momentum and impulse, and give their units.
3. Derive the relationship F = ma from the second law.
4. Calculate the recoil velocity of a gun.
5. State the principle of conservation of linear momentum.
6. Solve problems on elastic and inelastic collisions.

What are Newton’s laws of motion?

Newton’s first law

Newton’s first law of motion states that every body continues in its state of rest or of uniform motion in a straight line unless it is acted upon by a force.

The tendency of a body to remain at rest or, if moving, to continue its motion in a straight line is called inertia. That is why Newton’s first law is also called the law of inertia.

Newton’s second law

Newton’s second law of motion states that the rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction of the force.

F ∝ (mv – mu)/t

F ∝ m(v – u)/t

F ∝ ma

F = kma

Where k = 1,

F = ma

What is momentum?

The momentum of a body is the product of the mass and velocity of the body. The S.I. unit of momentum is kgm/s.

What is impulse?

Impulse is the product of force and time. It is also defined as the change in momentum. Thus both momentum and impulse have ‘Ns’ as a unit.

F = m(v – u)/t

Ft = mv – mu (where ‘mv – mu’ is the change of momentum)

F × t = I (Ns)

What is Newton’s third law?

Newton’s third law of motion states that to every action, there is an equal and opposite reaction.

A practical demonstration of this law is a bullet fired from a gun. The person holding the gun experiences a backward recoil force (the reaction). This force is equal to the propulsive force (the action) acting on the bullet.

According to Newton’s second law of motion, force is proportional to change in momentum. Therefore the momentum of the bullet is equal and opposite to the momentum of the gun, that is:

Mass of bullet × muzzle velocity = mass of gun × recoil velocity

Let m = mass of the bullet, v = velocity of the bullet, M = mass of the gun and V = velocity of the recoil of the gun. Then:

MV = mv

V = mv/M

What is conservation of linear momentum?

The principle of conservation of linear momentum states that when two or more bodies collide, their total momentum remains constant provided there is no external force acting on the system. This implies that in a closed or isolated system where there is no external force, the total momentum after collision remains constant. The principle is true for both elastic and inelastic collisions.

What are the types of collision?

There are two types of collision: elastic and inelastic.

Elastic collision

In an elastic collision, the two bodies collide and then move at different velocities. Both momentum and kinetic energy are conserved. Examples are collisions between gaseous particles and a ball which rebounds to its original height.

Suppose the two colliding bodies have masses m₁ and m₂, initial velocities u₁ and u₂, and final velocities v₁ and v₂. The conservation principle can be expressed mathematically as:

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Inelastic collision

In an inelastic collision, the two bodies join together after the collision and move with the same velocity. Momentum is conserved, but kinetic energy is not, because part of it is converted to heat or sound energy and leads to deformation.

Thus, the conservation principle can be rewritten as:

m₁u₁ + m₂u₂ = v(m₁ + m₂)

Momentum is a vector quantity, so all the velocities must be measured in the same direction. Give positive signs to the forward velocities and negative signs to the backward or opposite velocities.

Worked example: two bodies moving in the same direction

Two moving toys of masses 50 kg and 30 kg are travelling on the same plane with speeds of 5 m/s and 3 m/s respectively in the same direction. If they collide and stick together, calculate their common velocity.

M_AV_A + M_BV_B = V(M_A + M_B)

V = (M_AV_A + M_BV_B)/(M_A + M_B)

V = [(50 × 5) + (30 × 3)]/(50 + 30)

V = (250 + 90)/80

V = 340/80

V = 4.25 m/s

How do you solve a collision with bodies moving in opposite directions?

When body A and body B move towards each other and stick together after the collision, take the direction of M_A as positive. The velocity of M_B is then negative:

M_AV_A – M_BV_B = V(M_A + M_B)

V = (M_AV_A – M_BV_B)/(M_A + M_B)

Evaluation

1. State Newton’s first law of motion and define inertia.
2. Define momentum and impulse. Give the unit of each.
3. Write down the expression for the recoil velocity of a gun when a bullet is fired.
4. State two differences between an elastic and an inelastic collision.
5. What happens to the kinetic energy in an inelastic collision?

Assignment

1. Derive from Newton’s law the relationship between force, mass and acceleration.
2. State Newton’s laws of motion and explain the consequences of each law.
3. State the principle of conservation of linear momentum.
4. A 15 kg monkey hangs from a cord suspended from the ceiling of an elevator. The cord can withstand a tension of 200 N and breaks as the elevator accelerates. What was the elevator’s minimum acceleration (g = 10 m/s²)?