Elastic collision in one dimension

Any collision is elastic if the total kinetic energy of the colliding particles remains conserved. Let us consider two bodies A and B with masses m_1 and m_2 are moving with the initial velocity u_1 and u_2 respectively in the same direction and same straight line. In this problem let us suppose that velocity of one object is greater than other (u_1 > u_2) and they are on the collision path. In this situation object A will collide with B and this is called head on collision. After collision and according to our assumption velocity of A will decrease to v_1 and velocity of B will increase to v_2 If both objects are moving on the same direction after collision then we can say that
Total initial momentum of A and B before collision = m_1u_1 + m_2u_2
Total final momentum of A and B after collision = m_1v_1 + m_2v_2

According to conservation of momentum principle
m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2
m_1(v_1-u_1) = m_2(u_2-v_2) ——- (1)

Total kinetic energy of the particles before collision
KE_i = \frac{1}{2}m_1u_1^2 + \frac{1}{2}m_2u_2^2

Total kinetic energy of the particles after collision
KE_f = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2

For perfectly elastic collision KE_i = KE_f
KE_i = \frac{1}{2}m_1u_1^2 + \frac{1}{2}m_2u_2^2 = \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2
m_1(v_1^2-u_1^2) = m_2(u_2^2-v_2^2)
m_1(v_1-u_1)(v_1+u_1) = m_2(u_2-v_2)(u_2+v_2)
Dividing above equation by (1) we get
(u_1-u_2) = -(v_1 - v_2) ——- (2)

Here (u_1-u_2) is the relative velocity of approach of A towards B and (v_1-v_2) is the relative velocity of separation of B and A.

Equation (2) can also be written as
u_1+v_1 = u_2+v_2

Let us multiply above equation by m_2 and add equation (1) followed by some rearrangement we get
v_1 = \frac{m_1-m_2}{m_1+m_2}u_1 + \frac{2m_2}{m_1+m_2}u_2

Similarly let us multiply above equation by m_1 and subtracting equation (1) followed by some rearrangement we get
v_2 = \frac{2m_1}{m_1+m_2}u_1 + \frac{m_2-m_1}{m_1+m_2}u_2

Note 1: If mass of A is very very higher than B we can see that the velocity of A remains unchanged while that of body B changes after collision.
Note 2: If object A is much more smaller than B then velocity of A is changed but the velocity of B remains same.
Note 3: If mass of both body A and B are equal then the velocity if the particles are interchanged. v_1=u_2 and v_2=u_1

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