Physics 1D03 - Lecture 26 Collisions Conservation of Momentum Elastic and inelastic collisions

Physics 1D03 - Lecture 26 Collisions A collision is a brief interaction between two (or more) objects. We use the word “collision” when the interaction time Δt is short relative to the rest of the motion. 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. m1m1 m2m2 v 1,i v 2,i v 1,f v 2,f F1F1 F 2 = -F 1

Physics 1D03 - Lecture 26 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 (eg: billiard balls). If kinetic energy is lost (converted to other forms of energy), the collision is called inelastic (eg: tennis ball and a wall). A completely inelastic collision is one in which the two colliding objects stick together after the collision (eg: alien slime and a spaceship). Kinetic energy is lost in this collision.

Physics 1D03 - Lecture 26 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 is a vector equation. It applies to each component of p separately. m1m1 m2m2 v 1,i v 2,i v 1,f v 2,f

Physics 1D03 - Lecture 26 Elastic Collisions In one dimension (all motion along the x-axis): 1) Momentum is conserved: In one dimension, the velocities are represented by positive or negative numbers to indicate direction. 2) Kinetic Energy is conserved: We can solve for two variables if the other four are known.

Physics 1D03 - Lecture 26 One useful result: for elastic collisions, the magnitude of the relative velocity is the same before and after the collision: |v 1,i – v 2,i | = |v 1,f – v 2,f | (This is true for elastic collisions in 2 and 3 dimensions as well). An important case is a particle directed at a stationary target (v 2,i = 0): Equal masses: If m 1 = m 2, then v 1,f will be zero (1-D). If m 1 < m 2, then the incident particle recoils in the opposite direction. If m 1 > m 2, then both particles will move “forward” after the collision. before after

Physics 1D03 - Lecture 26 Elastic collisions, stationary target (v 2,i = 0): Two limiting cases: 1)If m 1 << m 2, the incident particle rebounds with nearly its original speed. v1v1 -v 1 2) If m 1 >> m 2, the target particle moves away with (nearly) twice the original speed of the incident particle. v1v1 v1v1 2v 1

Physics 1D03 - Lecture 26 Concept Quiz A tennis ball is placed on top of a basketball and both are dropped. The basketball hits the ground at speed v 0. What is the maximum speed at which the tennis ball can bounce upward from the basketball? (For “maximum” speed, assume the basketball is much more massive than the tennis ball, and both are elastic). a)v 0 b)2v 0 c)3v 0 ? v0 v0 v0 v0

Physics 1D03 - Lecture 26 Example A mad 60.0 kg physicist standing on a frozen lake throws a 0.5 kg stone to the east with a speed of 24.0 m/s. Find the recoil velocity of the mad physicist.

Physics 1D03 - Lecture 26 Example – inelastic collision: 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?

Physics 1D03 - Lecture 26 2 kg 4 kg 6 m/s 5 m/s An elastic collision: Two carts moving toward each other collide and bounce back. If cart 1 bounces back with v=2m/s, what is the final speed of cart 2 ?

Physics 1D03 - Lecture 26 Example A bullet of mass m is shot into a block of mass M that is at rest on the edge of a table. If the bullet embeds in the block, determine the velocity of the bullet if they land a distance of x from the base of the table, which has a height of h.