Concept:

Collisions are generally classified into two types: elastic and inelastic collisions.

This classification is based on the law of conservation, and the laws of collision state:

• Momentum is conserved in all collisions.
• Kinetic energy is also conserved in elastic collisions.
• In an inelastic collision, kinetic energy is not conserved. In a perfectly inelastic collision, the objects stick together after the collision.

Perfectly elastic collision

If the laws of momentum conservation and kinetic energy hold well during the collision.

That is, initial velocity = final velocity

Inelastic collision

If the law of momentum conservation holds well during the collision while kinetic energy does not.

However, inelastic collisions follow the law of conservation of linear momentum.

Coefficient of restitution (e)

\(e = \frac{{Relative\;velocity\;after\;collision}}{{Relative\;velocity\;before\;collision}} = \frac{{{v_2} - {v_1}}}{{{u_1} - {u_2}}}\)

• For a perfectly elastic collision, e = 1
• For an inelastic collision, e <1
• For a perfectly inelastic collision, e = 0

Calculation:

Let us assume that,

Mass of body A, m_A = 2 kg

Block A, u_A = 4 m/s . velocity

Mass of body B, m_B = 3 kg

Velocity of block B, u_B = -2 m/s

Now, according to the law of conservation of momentum

m_A u_A + m_B u_B = m_A v_A + m_B v_B

2 x 4 - 3 x (-2) = 2 x v_A + 3 x v_B

2 x v_A + 3 x v_B = 2      -----(1)

Now, for an inelastic collision, the coefficient of restitution is always one and can be expressed as

\(e=1=\frac{{{v}_{B}}-{{v}_{A}}}{{{u}_{A}}-{{u}_{B}}}=\frac{{{v}_{B}}-{{v}_{A}}}{4+2}\Rightarrow {{v}_{B}}-{{v}_{A}}=6\)     ------(2)

Thus, comparing both quadratic equations, we obtain the values of the velocities of bodies A and B as follows

\({{v}_{B}}=\frac{14}{5}m/s\)

And now substituting this value into eq (2), we get, \({{v}_{A}}=-\frac{16}{5}m/s\)

Thus, we can see that body A will be moving in the opposite direction compared to body B.

Therefore, in this case, body A and body B will move in opposite directions in an elastic collision.