Lessons 15–16 · 2 lessons · N. Sh. Turdiyev, K. A. Tursunmetov, A. G. Ganiyev, K. T. Suyarov, J. E. Usarov, A. K. Avliyoqulov. Physics Grade 10, 1st edition. Niso Poligraf Publishing House, Tashkent, 2017
16
Perfectly elastic and inelastic collisions of bodies
Textbook: pp. 53–56
GoalUse conservation of momentum and energy in perfectly elastic and inelastic collisions.
A collision is the interaction of two or more bodies lasting a very short time. The momentum of a body is p = mυ (a vector), and in a closed system the vector sum of the momenta is conserved. In a perfectly inelastic collision the bodies deform and move together (with the same velocity): m₁υ₁ + m₂υ₂ = (m₁ + m₂)u, so u = (m₁υ₁ + m₂υ₂)/(m₁ + m₂) (velocities taken along one axis with their signs). Part of the mechanical energy turns into internal energy (heat, deformation), so kinetic energy is not conserved. In a perfectly elastic collision the deformation leaves no residue and both momentum and kinetic energy are conserved: m₁υ₁ + m₂υ₂ = m₁υ₁′ + m₂υ₂′ and m₁υ₁²/2 + m₂υ₂²/2 = m₁υ₁′²/2 + m₂υ₂′²/2. If the second body is at rest (υ₂ = 0), υ₁′ = (m₁ – m₂)υ₁/(m₁ + m₂) and υ₂′ = 2m₁υ₁/(m₁ + m₂). If the masses are equal the bodies exchange velocities. Real collisions lie between these two ideal cases.
Worked examples
A plasticine ball (2 kg, 6 m/s) sticks to a 4 kg ball at rest: u = 2·6/(2 + 4) = 2 m/s. Kinetic energy was 2·36/2 = 36 J and becomes 6·4/2 = 12 J: 24 J turned into internal energy.
Elastic collision: a 2 kg ball at 6 m/s hits a 1 kg ball at rest. υ₁′ = (2 – 1)·6/3 = 2 m/s, υ₂′ = 2·2·6/3 = 8 m/s. Check: momentum 12 = 2·2 + 1·8 = 12; energy 36 = 4 + 32 = 36.
Class activity
Only with the teacher: hang two identical metal balls on strings as a “Newton’s cradle”; pull one aside and release it and observe how the ball at the other end jumps. Explain with conservation of momentum and energy. Keep fingers out of the way of the balls.
Practice
1
How do perfectly elastic and perfectly inelastic collisions differ?
In an elastic one both momentum and kinetic energy are conserved; in an inelastic one the bodies move together and part of the kinetic energy becomes internal energy.
2
A 20 t wagon moving at 4 m/s hits a 20 t wagon at rest and they couple. What is their common speed (m/s)?
2
3
A 2 kg ball at 6 m/s hits a 1 kg ball at rest elastically. What is the second ball’s speed (m/s)?
8
4
Why is kinetic energy not conserved but momentum is conserved in an inelastic collision?
External forces are negligible, so momentum is conserved; part of the energy goes into deformation and heat.