Momentum and kinetic energy in interactions
Section: Dynamics | Syllabus: Cambridge AS Level Physics (9702)
Elastic and Inelastic Interactions
While momentum is always conserved in interactions, the total kinetic energy (E_k) may not be. Kinetic energy is calculated as E_k = (1/2)mv^2.
Definitions- Elastic Interaction: Total kinetic energy is conserved (no energy is transferred to other forms).
- Inelastic Interaction: Total kinetic energy is not conserved. Some energy is transferred into thermal energy (heat), sound, or elastic potential energy.
A 0.16 kg white ball at 1.0 m s^-1 hits a stationary 0.16 kg red ball. If the white ball stops and the red ball moves off at 1.0 m s^-1:
KE CalculationTotal E_k, before = (1/2)(0.16)(1.0)^2 = 0.08 J
Total E_k, after = (1/2)(0.16)(1.0)^2 = 0.08 J
Since E_k is conserved, the collision is perfectly elastic.
Worked Example: Railway Wagons (Inelastic)A 1900 kg wagon at 3 m s^-1 hits a stationary 2200 kg wagon. They join and move together. Is it elastic?
Solution- Find Velocity (v): 1900 × 3 = (1900 + 2200)v v ≈ 1.4 m s^-1
- Initial KE: (1/2)(1900)(3)^2 = 8550 J
- Final KE: (1/2)(4100)(1.4)^2 ≈ 4018 J
Significant KE is lost (8550 → 4018), so the collision is inelastic.
Relative Speed of Approach and Separation
Relative speed is the speed of one object compared to another. It provides a quick way to test for elasticity.
Perfectly Elastic Collision:
Relative Speed of Approach = Relative Speed of Separation
Calculating Relative Speed:
- Opposite Directions: Sum of speeds (v_A + v_B).
- Same Direction: Difference of speeds (|v_A - v_B|).
- One Stationary: Just the speed of the moving object.
Show two cars on a road: (1) approaching each other (large relative speed) and (2) one overtaking the other (small relative speed).
Worked Example: Repulsion without ContactAn alpha particle (m) at v_1 approaches a stationary helium atom (m). They repel via electrostatic forces. If they don't touch, is it elastic?
AnalysisInteractions that involve non-contact forces (like electric repulsion or magnets) are often perfectly elastic because no energy is lost to heat or permanent deformation.
Explosions and Energy
In physics, a separation or explosion is the reverse of a collision. Separations are always inelastic.
The kinetic energy gained by the moving parts comes from stored potential energy (e.g., chemical energy in gunpowder or elastic potential in a compressed spring).
Worked Example: Gun Recoil EnergeticsA 3 kg gun fires a 5 g bullet at 400 m s^-1. Initial KE is zero.
SolutionTotal E_k, after = E_k, bullet + E_k, gun
E_k, bullet = (1/2)(0.005)(400)^2 = 400 J
E_k, after ≈ 400 J (neglecting small gun recoil KE).
Since KE increased from 0 to 400 J, energy was transferred from chemical stores. This is inelastic.
Core Summary
- Elastic: KE conserved, Relative speed preserved.
- Inelastic: KE lost to other forms, objects often stick together.
- Total Energy: Always conserved throughout all interactions.
Interactive revision notes, videos and practice questions load below.