Elastic and Inelastic Collisions: MDCAT Physics notes

Types of collision

In every collision of an isolated system, momentum is conserved and total energy is conserved. What distinguishes the types is the kinetic energy.

A perfectly (completely) inelastic collision is one in which the bodies stick together and move with a common velocity; maximum KE is lost.

Elastic collision in one dimension

Conserving both momentum and KE for a head-on collision gives the important result:

$$u_1 - u_2 = -(v_1 - v_2)$$

The relative speed of approach equals the relative speed of separation, regardless of the masses. If 2 kg at $5\ \text{m s}^{-1}$ catches 3 kg at $1\ \text{m s}^{-1}$, they approach at $4\ \text{m s}^{-1}$ and separate at $4\ \text{m s}^{-1}$.

Final velocities (for $m_2$ initially at rest):

$$v_1 = \frac{m_1 - m_2}{m_1 + m_2}u_1, \qquad v_2 = \frac{2m_1}{m_1+m_2}u_1$$

Special cases

• Equal masses: the bodies exchange velocities ($v_1 = u_2$, $v_2 = u_1$). A moving ball hitting an identical ball at rest stops, and the second moves off with the first one's speed.
• Light body hits a very heavy body at rest: it rebounds with nearly the same speed; the heavy body barely moves.
• Heavy body hits a light body at rest: heavy body continues almost unchanged; light body moves off at nearly $2u_1$.

Bodies that stick together

$$v = \frac{m_1u_1 + m_2u_2}{m_1 + m_2}$$

• 2 kg at $3\ \text{m s}^{-1}$ sticks to 1 kg at rest: $v = 6/3 = 2\ \text{m s}^{-1}$.
• Mass dropped vertically into a moving wagon: it brings no horizontal momentum, so $1000 \times 50 = 1250\,v$, $v = 40\ \text{km h}^{-1}$.
• Momentum is a vector: if two equal momenta at right angles combine, the composite moves at $45^\circ$ to each.

Zero total momentum

Two equal masses approaching each other with equal speeds have total momentum $mv + (-mv) = 0$ before collision, and so zero after as well, whatever the type of collision.

Key formulas

• $m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$
• Elastic: $\tfrac12 m_1u_1^2 + \tfrac12 m_2u_2^2 = \tfrac12 m_1v_1^2 + \tfrac12 m_2v_2^2$
• Elastic: $u_1 - u_2 = v_2 - v_1$
• Sticking: $v = (m_1u_1+m_2u_2)/(m_1+m_2)$

Common MDCAT traps

• Momentum is conserved in all collisions; only KE separates elastic from inelastic.
• In a crash (inelastic), KE is not conserved but momentum is.
• Relative speed of separation equals approach in elastic collisions; it does not depend on masses.
• Total energy is conserved even in inelastic collisions; only KE is lost.

Quick revision

• Elastic: momentum + KE conserved.
• Perfectly inelastic: bodies stick together.
• Equal masses, elastic, head-on: velocities exchange.
• Equal and opposite momenta: total zero before and after.