Conservation of Linear Momentum

A Closed / Isolated System:
A closed or isolated system means a system in which no external force acts. Or it is a system in which there is no interference by any external factor.

Different Cases Of Collision:

If masses, M₁, and M₂, moving in the same direction with initial velocities U₁ and U₂, respectively, collide with each other and move in different directions

From conservation of linear momentum, the total momentum before collision is equal to the total momentum after collision; therefore,

M₁U₁ + M₂U₂ = M₁V₁ + M₂V₂

When two bodies moving in opposite directions collide,

M₁U₁ + M₂U₂ = M₁V₁ + M₂V₂

U₂ is negative because it is moving in the opposite direction.

Therefore:

M₁U₁ + M₂(-U₂) = M₁V₁ + M₂V₂

M₁U₁ – M₂U₂ = M₁V₁ + M₂V₂

When a stationary body collides with a moving body, and move apart.

M₁U₁ + M₂U₂ = M₁V₁ + M₂V₂

Remember that U₂ = 0 m/s (the body is at rest / stationary)
Then we will get 

M₁U₁ + M₂(0) = M₁V₁ + M₂V₂

M₁U₁ + 0 = M₁V₁ + M₂V₂

M₁U₁ = M₁V₁ + M₂V₂

If two bodies collide with each other, and stick together, and move with common velocity.

M₁U₁ + M₂U₂ = (M₁ + M₂)V

V is the common velocity of the body.

Example 1

A ball of mass 0.5 kg moving at 10 ms^-1 collides with another ball of equal mass at rest. If the two balls move off together after the impact, calculate their common velocity.

Solution

M₁ = M₂ = 0.5 kg
U₁ = 10 ms^-1, U₂ = 0 ms^-1
V₁ = V₂ = V

using the formula;

M₁U₁ + M₂U₂ = M₁V₁ + M₂V₂

M₁U₁ + M₂(0) = M₁V₁ + M₂V₂

but V₁ = V₂ = V

M₁U₁ = (M₁ + M₂)V

Substitute the values given in question

0.5(10) = (0.5 + 0.5)V

0.5 × 10 = 1 × V

∴ V = 5 m/s

Example 2

An object A of mass 10 kg moving with a velocity of 3 m/s collides with an object B of mass 15 kg moving with a velocity of 4 m/s in opposite direction. If A and B stick together after collision. Calculate their common velocity?

Solution

M₁ = 10 kg, M₂ = 15 kg
U₁ = 3 m/s, U₂ = -4 m/s (opposite direction)
Common velocity, V = ?

using the formula;

M₁U₁ – M₂U₂ = M₁V₁ + M₂V₂

M₁U₁ – M₂U₂ = (M₁ + M₂)V

(10 × 3) – (15 × 4) = (10 + 15)V

30 – 60 = 25V

-30 = 25V

V = \( \frac{-30}{25} \)

V = -1.2 m/s

The negative sign implies that the two objects collide and move in the same direction as B.

Example 3

A body P of mass 8 kg moving with a velocity of 30 m/s collides with another body Q moving with velocity of 15m/s in the opposite direction. If both bodies now move in the direction of P at a velocity of 7.5 m/s, calculate the mass of Q

Solution

M₁ = 8 kg, M₂ = ?
U₁ = 30 m/s, U₂ = -15 m/s (opposite direction)
Common Velocity V = 7.5 m/s

using the formula;

M₁U₁ – M₂U₂ = M₁V₁ + M₂V₂

M₁U₁ – M₂U₂ = (M₁ + M₂)V

(8 × 30) – (M₂ × 15) = (8 + M₂) × 7.5

240 – 15M₂ = 60M₂

240 = 75M₂

M₂ = \( \frac{240}{75} \)

M₂ = 3.2kg