Collisions Conservation of Momentum Elastic and inelastic collisions Serway 9.3-9.4 For practice: Chapter 9, problems 10, 11, 23, 70, 75
Momentum: p = mv Impulse (a vector) is defined as F t (for a constant force), or F dt in general. Newton s Second Law again: F = dp dt or, alternatively, (change in p) = (total impulse from external forces)
Newton #3 and Momentum Conservation Two particles interact: p 1 = F 1 t p 2 = F 2 t = -F 1 t = - p 1 F 1 m 1 F 2 = -F 1 m 2 The momentum changes are equal and opposite. The total momentum of the system (two particles) doesn t change: p = p 1 + p 2 ; p = p 1 + p 2 = 0 Total momentum p is constant The fine print: only internal forces act. External forces would transfer momentum into or out of the system.
Momentum Conservation Newton s 3 rd Law: F 2 = -F 1 leads to the F 1 Law of Conservation of Momentum: m 1 F 2 = -F 1 m 2 The total momentum (vector sum) of all the particles in a system remains constant if there are no external forces acting on the system.
Example: roller basketball A (stationary) 100-kg basketball player on roller skates shoots a 1-kg basketball horizontally at a speed of 10 m/s. What happens (if there is no friction)? Is momentum still conserved after an opponent catches the ball and holds it? What happens if the ball is thrown upward at an angle of 37 degrees to the floor?
v Quiz π + 2v π A kaon ( a strange type of subatomic particle) at rest spontaneously splits into three other particles, called π +, π, and π 0, which fly off in different directions. The π + and π have equal masses, but the π 0 mass is different. The velocities of two particles are shown; in which direction is the third one moving? A) B) C) D) E) We can t tell exactly without knowing its mass
Collisions A collision is a brief interaction between two (or more) objects. m 1 v 1,i v 2,i m2 During a collision, the objects exert equal and opposite forces on each other. We assume these internal forces are much larger than any external forces on the system. We can ignore external forces if we compare velocities just before and just after the collision, and if the interaction force is much larger than any external force. F 1 F 2 = -F 1 v 1,f v 2,f
If there are no external forces, then the total momentum is conserved: p 1,i + p 2,i = p 1,f + p 2,f This will be a good approximation if the impulse from any external forces is small. This is a vector equation. It applies to each component of p separately.
Example: A neutron, with mass m = 1 amu (atomic mass unit), travelling at speed v 0, strikes a stationary deuterium nucleus (mass 2 amu), and sticks to it, forming a nucleus of tritium. What is the final speed of the tritium nucleus? What does the energy balance look like?
Elastic and Inelastic Collisions Momentum is conserved in collisions. Kinetic energy is sometimes conserved; it depends on the nature of the interaction force: A collision is called elastic if the total kinetic energy is the same before and after the collision. If the interaction force is conservative, a collision between particles will be elastic If kinetic energy is lost (converted to other forms of energy), the collision is called inelastic. A completely inelastic collision is one in which the two colliding particles stick together after the collision. Question: Why not define completely inelastic to mean the final kinetic energy is zero?
Quiz When two stars moving through the Galaxy pass close to each other, their mutual gravitational pull causes them to change speed and direction. We can call this a collision, even though there was no contact between the stars. Will this be an elastic or inelastic collision? a) elastic or nearly elastic b) Inelastic
An elastic collision in slow motion: If k = 1200 N/m, what is the maximum compression of the spring during the collision? 6 m/s 3 m/s 2 kg 4 kg Is the momentum constant at all times? Is the kinetic energy constant at all times? Is the mechanical energy constant at all times?
4 kg 4 kg QUIZ (no friction, ideal spring, ) Which quantities do not remain constant at all times during the collision? 4 kg A) Momentum only B) Kinetic energy only C) Mechanical energy only D) All are constant E) Two are not constant 4 kg 4 kg
An elastic collision in slow motion: If k = 1200 N/m, what is the maximum compression of the spring during the collision? 6 m/s 3 m/s 2 kg 4 kg