Law of Conservation of Momentum & Derivation

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 in the same direction. 

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. 

Derivation of the Law 

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 

We know the formula of momentum, 

So, Initial momentum (A) = m_au_a 

& Initial momentum (B) = m_bu_b 

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_a v_b 

The force on A due to B is applied and thus an equal and opposite force will act on B due to A according to Newton’s third law of motion. Thus,  

F_ab = -F_ba 

Since F = mass x acceleration  

Thus, m_ba_b = -m_aa_a 

Since a = (v – u )/ t, so 

m_b ( v_b – u_b ) = -m_a (v_a – u_a ) 

m_bv_b – m_b u_b = -m_a v_a + m_a u_a 

m_bv_b + m_a v_a = m_a u_a+ m_b u_b 

Such that,  

Magnitude of the final momenta = magnitude of the initial momenta  

Questions and answers 

Question 1: There are two balls of masses 3 kg and 4 kg respectively moving in the same direction. 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 of the balls 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: 

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) = 54 kgm/s 

Final momentum = 3 (u₁) + 4 (5) = (3u₁ + 20) kgm/s 

Equating both, 

3 u₁ + 20 = 54 

  u₁ =

343343

= 11.3 m/s 

Velocity of ball after collision is 11.3 m/s 

Question 2: There are two cars, A and B. Car A of mass 1500 kg, traveling at 30 m/s collides with car B of mass 3000 kg traveling at 20 m/s in the same direction. 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 = 20 m/s 

Initial momentum, = m_a u_a + m_b u_b    

                                     = 1500 (30) + 3000 (20)  

                                     = 105000 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)  

                                   = 60000 + 30000 v 

According to law of conversation of momentum. 

Initial momentum = Final momentum  

                    105000 = 60000 + 30000 v 

                ⇒ 30000 v = 45000 

v  = 

45000300004500030000

= 1.5 m/s. 

Velocity of car B after collision is 1.5 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