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 = fractionmv^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. Worked Example: Snooker Ball (Elastic) 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 Calculation Total E_k, before = fraction(0.16)(1.0)^2 = 0.08 J Total E_k, after = fraction(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: fraction(1900)(3)^2 = 8550 J Final KE: fraction(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. Diagram: Relative Speed Visualization 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 Contact An 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? Analysis Interactions 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 Energetics A 3 kg gun fires a 5 g bullet at 400 m s^-1. Initial KE is zero. Solution Total E_k, after = E_k, bullet + E_k, gun E_k, bullet = fraction(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.
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