Conservation and Momentum

The Law of Conservation of Momentum

1. According to Newton’s 3rd Law, the force acting on object X by object Y (F_x) …………… is equal to the force acting on object Y by object X (F_y).

2. As per Newton’s 2nd Law, since force is equal to the rate of change of momentum, the rate of change of momentum is the same for both objects.

3. Since the objects have opposite momentum directions, resulting in different signs, the net change in momentum is zero.

Consider the following scenario:

Let’s take a simple example to illustrate this law. Imagine two billiard balls, one with a mass of 1 kg and the other with a mass of 2 kg, are initially at rest on a frictionless table. The first ball is then struck by a cue stick, causing it to move with a velocity of 4 m/s towards the second ball. The collision between the two balls is perfectly elastic, which means that no energy is lost during the interaction.

According to the law of conservation of momentum, the total momentum of the system before and after the interaction must be the same. 

Before the collision, the momentum of the first ball is:

P₁ = m₁v₁ = 1 kg x 4 m/s = 4 kg m/s

The momentum of the second ball is zero, since it is at rest:

P₂ = m₂v₂ = 2 kg x 0 m/s = 0 kg m/s

Therefore, the total momentum of the system before the collision is:

P₁ + P₂ = 4 kg m/s + 0 kg m/s = 4 kg m/s

After the collision, the first ball rebounds with a velocity of 2 m/s in the opposite direction, while the second ball moves with a velocity of 2 m/s in the original direction.

The momentum of the first ball after the collision is:

p1′ = m1v1′ = 1 kg x (-2 m/s) = -2 kg m/s

The momentum of the second ball after the collision is:

p2′ = m2v2′ = 2 kg x 2 m/s = 4 kg m/s

Therefore, the total momentum of the system after the collision is:

p1′ + p2′ = (-2 kg m/s) + 4 kg m/s = 2 kg m/s

As we can see, the total momentum of the system before and after the collision is the same (4 kg m/s and 2 kg m/s, respectively), which verifies the law of conservation of momentum.

Consider the following scenario:

Suppose there are two ice skaters, one with a mass of 70 kg and the other with a mass of 50 kg, gliding on a frictionless ice rink towards each other. The skater with the mass of 70 kg is skating at a velocity of 5 m/s towards the skater with the mass of 50 kg, who is skating at a velocity of 8 m/s towards the skater with the mass of 70 kg.

When the two skaters collide, they stick together and move off in the same direction. According to the law of conservation of momentum, the total momentum of the system before and after the collision must be the same.

Before the collision, the momentum of the first skater is:

P₁ = m₁v₁ = 70 kg x 5 m/s = 350 kg m/s

The momentum of the second skater is:

P₂ = m₂v₂ = 50 kg x (-8 m/s) = -400 kg m/s (negative because the skater is moving in the opposite direction)

Therefore, the total momentum of the system before the collision is:

P₁ + P₂ = 350 kg m/s – 400 kg m/s = -50 kg m/s

After the collision, the two skaters stick together and move off in the same direction. Let’s call their combined mass M and their velocity after the collision v:

M = m₁ + m₂ = 70 kg + 50 kg = 120 kg

v = (m₁v₁ + m₂v₂) / (m₁ + m₂) = (70 kg x 5 m/s + 50 kg x (-8 m/s)) / 120 kg = -0.83 m/s

Therefore, the momentum of the system after the collision is:

p’ = Mv = 120 kg x (-0.83 m/s) = -100 kg m/s

As we can see, the total momentum of the system before the collision (-50 kg m/s) is equal to the total momentum of the system after the collision (-100 kg m/s), which verifies the law of conservation of momentum.

Energy changes during collisions:

When considering the change in kinetic energy during a collision, it’s important to remember that energy is not lost but rather transformed into other forms such as heat, sound, and permanent material distortion. This distortion causes an increase in the internal potential energy of the bodies involved.

If there is no loss of kinetic energy (K.E. = ½ mv²), then the collision is considered perfectly elastic. However, if kinetic energy is lost, the collision is considered inelastic. When all of the kinetic energy is lost, the collision is considered completely inelastic, which occurs when the two colliding bodies stick together on impact and have zero combined velocity.

According to the equation mentioned in the previous section, when a force acts on a body over a period of time, it results in a change in momentum. When two bodies collide, they exert a force over the same period of time, resulting in a change in momentum. By Newton’s third law, if object A exerts a force on object B, then object B exerts an equal and opposite force on object A. Therefore, the change in momentum is equal and opposite.

The principle of conservation of momentum states that the total momentum of a system of colliding bodies, before and after the collision, is conserved

provided that there are no external forces acting on the system. This principle also applies to explosions, where objects move apart. For instance, a rocket gains momentum by the controlled explosion of fuel, where the momentum of the hot exhaust gases equals the momentum of the rocket.