Elastic and inelastic collisions
Newton's Laws & Momentum - OCR A-Level Physics
Case 1 : Elastic collision
Elastic collision
A collision in which the total kinetic energyThe energy an object has due to its motion. $E_k = \frac{1}{2} m v^2$. Measured in joules. after is equal to the total kinetic energy before. Both momentum and kinetic energy are conserved. No KE is converted into heat, sound or deformation. Approximated by collisions between very hard objects and by collisions between subatomic particles.
A collision in which the total kinetic energyThe energy an object has due to its motion. $E_k = \frac{1}{2} m v^2$. Measured in joules. after is equal to the total kinetic energy before. Both momentum and kinetic energy are conserved. No KE is converted into heat, sound or deformation. Approximated by collisions between very hard objects and by collisions between subatomic particles.
Case 2 : Inelastic collision
Inelastic collision
A collision in which the total kinetic energy after is less than the total kinetic energy before. Momentum is still conserved. Some KE is converted to heat, sound or permanent deformation of the colliding objects. Most everyday collisions are inelastic.
A collision in which the total kinetic energy after is less than the total kinetic energy before. Momentum is still conserved. Some KE is converted to heat, sound or permanent deformation of the colliding objects. Most everyday collisions are inelastic.
Case 3 : Perfectly inelastic collision
Perfectly inelastic collision
The objects stick together on impact and move as one body afterwards. This is the case where the MAXIMUM amount of kinetic energy is lost while still conserving momentum.
The objects stick together on impact and move as one body afterwards. This is the case where the MAXIMUM amount of kinetic energy is lost while still conserving momentum.
$$\tfrac{1}{2} m_1 u_1^{2} + \tfrac{1}{2} m_2 u_2^{2} \stackrel{?}{=} \tfrac{1}{2} m_1 v_1^{2} + \tfrac{1}{2} m_2 v_2^{2}$$
- Momentum is conserved in ALL three cases. Total energy is also conserved if you account for heat, sound and deformation. It is the KINETIC energy total that can drop in cases 2 and 3.
- To test if a collision is elastic: compute total KE before and total KE after. If they are equal, the collision is elastic.
- The KE lost in an inelastic collision equals $\Delta E_k = E_{k,\text{before}} - E_{k,\text{after}}$. State where it goes (heat, sound, deformation).
- For two objects of equal mass where one is initially at rest, an elastic collision causes a complete velocity swap: the moving one stops, the stationary one moves off with the original velocity.
Common Mistake
HIGH
Wrong: Writing that momentum is not conserved in an inelastic collision.
Right: Momentum is ALWAYS conserved in every collision (elastic, inelastic and perfectly inelastic), provided no external resultant force acts. It is KINETIC ENERGY that is not conserved in an inelastic collision. The lost KE is transferred to thermal, sound and deformation stores.
Right: Momentum is ALWAYS conserved in every collision (elastic, inelastic and perfectly inelastic), provided no external resultant force acts. It is KINETIC ENERGY that is not conserved in an inelastic collision. The lost KE is transferred to thermal, sound and deformation stores.
Worked Example [5 marks]
A 2.0 kg trolley moving at 3.0 $\text{m s}^{-1}$ collides with a stationary 1.0 kg trolley. After the collision they stick together. Calculate (a) the velocity after the collision, (b) the kinetic energy lost.
Show Solution
1
Apply conservation of momentum (objects stick, so Case 2 form).
$m_1 u_1 + m_2 u_2 = (m_1 + m_2) v$
[1]2
$(2.0)(3.0) + (1.0)(0) = (3.0) v \;\Rightarrow\; v = 2.0 \; \text{m s}^{-1}$
[1]3
$E_{k,\text{before}} = \tfrac{1}{2} (2.0)(3.0)^{2} = 9.0 \; \text{J}$
[1]4
$E_{k,\text{after}} = \tfrac{1}{2} (3.0)(2.0)^{2} = 6.0 \; \text{J}$
[1]5
$\Delta E_k = 9.0 - 6.0 = 3.0 \; \text{J}$ lost to heat, sound and deformation.
[1]Answer
(a) $v = 2.0 \; \text{m s}^{-1}$. (b) 3.0 J of KE lost.
Examiner Tips and Tricks
- "Show that this collision is inelastic": compute total KE before and after; quote both; then conclude KE has decreased so the collision is inelastic.
- Never write "KE is lost". KE is TRANSFERRED to other energy stores. Saying "lost" can lose a marking point in extended-response questions.
- If a collision is stated to be elastic, you have TWO equations available: conservation of momentum AND conservation of KE. Two equations, two unknowns; you can solve for both final velocities.
Before/after sketch of a 2.0 kg trolley striking a 1.0 kg trolley: arrow showing 3.0 $\text{m s}^{-1}$ before, single combined trolley with arrow showing 2.0 $\text{m s}^{-1}$ after.