Law of Conservation of Momentum

Key Concepts

• Conservation of momentum

Introduction: 

In our daily life we make many observations such as a fast bowler taking a run up before bowling, a tennis player moving her racket backwards before hitting the tennis ball and a batsman moving his bat backwards before hitting the cricket ball. All these activities are performed in order to make the ball move with a great speed when hit/thrown. The balls would rather move slowly if these activities are not done. But did you know in all of these the momentum is conserved. In this session we are going to discuss about law of conservation of momentum in objects moving towards each other and when one object is at rest. 

Explanation: 

Momentum 

Momentum is a vector quantity, as velocity, in its formula, is a vector quantity. It is directed along the velocity of the moving object. 

Momentum, p α m and v 

This means,       p = mv 

The S I unit of mass is kg and of velocity is m/s. 

Therefore, the S I unit of p = S I unit of mass x S I unit of velocity. 

                                                   =  kg * m/s 

                                                   =  kg.m/s 

Thus, the S I unit of momentum is kg.m/s. 

Law of Conservation of Momentum 

The sum of momenta of the two objects before collision is equal to the sum of momenta after the collision, provided there is no external unbalanced force acting on them.  

This is known as the law of conservation of momentum. 

Application of Law 

1. When moving towards each other: 

Consider an object A has mass ‘m_a,’ and object B has mass ‘m_b.’ 

The initial velocity of an object A is u_a 

and initial velocity of an object B is u_b = 0 m/s 

We know the formula of momentum, 

So, Initial momentum (A) = m_a u_a 

& Initial momentum (B) = -m_b u_b = 0 m/s 

Both collide such that,  

The final velocity of an object A is (-v_a) 

and final velocity of an object B is v_b 

Final momentum after the collision:  

Final momentum of an object A is (-m_a v_a) 

Final momentum of an object B is m_b v_b 

According to law of conservation,  

Magnitude of the final momenta = magnitude of the initial momenta 

Such that the equation becomes, 

-m_a v_a + m_b  v_b = m_a u_a 

2. When one object is stationary and other collides with it 

Consider an object A has mass ‘m_a,’ and object B has mass ‘m_b.’ 

The initial velocity of an object A is u_a 

and initial velocity of an object B is u_b = 0 m/s 

We know the formula of momentum, 

So, Initial momentum (A) = m_a u_a 

& Initial momentum (B) = m_b u_b = 0 m/s 

Both collide such that,  

The final velocity of an object A is -v_a  

and final velocity of an object B is v_b 

Final momentum after the collision:  

Final momentum of an object A is (-m_a v_a) 

Final momentum of an object B is m_b v_b 

According to law of conservation,  

Magnitude of the final momenta = magnitude of the initial momenta 

Such that the equation becomes, 

-m_a v_a + m_b  v_b = m_a u_a 

Questions and answers 

Question 1: There are two balls of masses 3 kg and 4 kg respectively moving towards each other. 3Kg ball is moving with a speed of 10 m/s and 4 kg ball is moving with a speed of 6 m/s.   
After collision the 2nd ball of mass 4 kg moves with velocity 5 m/s. Find the velocity of the ball at mass 3 kg with respect to ground. 

Answer:

Mass of ball , m₁ = 3 kg  

                          m₂ = 4 kg 

Initial speed of the ball, u₁ = 10 m/s 

                                              u₂ = -6 m/s 

Final speed of the ball, v₁ = ? 

                                            v₂ = 5 m/s 

According to law of conversation of momentum. 

Initial momentum = 3 (10) + 4 (-6) = 6 kgm/s 

Final momentum = 3 (v) + 4 (5) = (3v + 20) kgm/s 

Equating both, 

3v + 20 = 6 

3v = -14 

v =

−143−143

= -3.66 m/s 

Velocity of ball after collision is -3.66 m/s. 

Question 2: There are two cars, A and B. Car A of mass 1500 kg, traveling at 30 m/s collide with car B of mass 3000 kg at rest. After collision, the velocity of car A becomes 40 m/s. Calculate the velocity of car B after the collision?

Answer:

Mass of car A = 1500 kg 

Mass of car B = 3000 kg 

Before collision, 

Initial velocity , u_A = 30 m/s 

                               u_B = 0 m/s 

Initial momentum, = m_a u_a + m_b u_b    

                                    = 1500 (30) + 3000 (0)  

                                    = 45,000 kgm/s 

After collision, 

Final velocity , v_A = -40 m/s 

                             v_B = +v 

Final momentum,  = m_A v_A + m_B v_B    

                                    = 1500 (-40) + 3000 (+v)  

                                    = -60,000 + 3000 v 

According to law of conversation of momentum. 

Initial momentum = Final momentum  

                      45000 = -60000 + 3000 v 

                 = 3000 v = 105000 

v = 105000/3000 = 35 m/s 

Velocity of car B after collision is 35 m/s. 

Summary

• Law of conservation of momentum- “The sum of momenta of the two objects before collision is equal to the sum of momenta after the collision provided there is no external unbalanced force acting on them.
• m₁u₁+ m₂u₂ = m₁v₁+ m₂V₂