Understanding Momentum and Collisions in Physics

1. Momentum, Impulse and Collisions • Momentum • everyday connotations? • physical meaning • the “true” measure of motion (what changes in response to applied forces) • Momentum (specifically Linear Momentum) defined • to be generalized later • so • note: momentum is a vector • px = mvx , py = mvy , pz = mvz

2. SFx (Fav)x t • Impulse • During a constant force Integral is area under the curve!

3. For a particle initially at rest • the particle’s momentum equals the impulse that accelerated the particle from rest to its current state of motion. • (analogous to K = work to accelerate particle from rest…) • Kinetic energy can be written in terms of momentum and mass

4. Consider 2 particles of the same mass, the second with twice the speed of the first. • How do their momenta compare? • How do their Kinetic Energies compare? • Consider 2 particles with the same speed, the second with twice the mass of the first. • How do their momenta compare? • How do their Kinetic Energies compare? • Consider 2 particles initially at rest with the same size force acting for the same amount of time, the second with twice the mass of the first. • How do their momenta compare? • How do their Kinetic Energies compare? • Consider 2 particles initially at rest with the same size force acting through the same distance, the second with twice the mass of the first. • How do their momenta compare? • How do their Kinetic Energies compare?

5. Ex: A 0.40 kg ball impacts a wall horizontally with a speed of 30 m/s and rebounds horizontally with a speed of 20 m/s. The ball is in contact with the wall for .01s. Determine the impulse and average force on the ball. Ex: A 0.40 kg soccer ball traveling horizontally to the left at 20 m/s is kicked up and to the right at a 45 degree angle with a speed of 30 m/s. The ball is in contact with the foot of the kicker for .01s. Determine the impulse and average force on the ball.

6. Conservation of Momentum: an application of action-reaction • 2 interacting objects, no external forces • Generalize and consider external forces: • If the vector sum of external forces on a system is zero, the total momentum of the system is constant.

7. mAvA1 + mBvB1 = mAvA2 + mBvB2 • Using conservation of momentum in problems: • determine if momentum is conserved • select a coordinate system (momentum is a vector!) • sketch before and after diagrams • relate total initial momentum to total final momentum, component by component! • solve equations (use additional equations as appropriate such as conservation of energy)

8. Example: Rifle recoil: A 3 kg rifle is used to fire a 5 g bullet. The velocity of the bullet relative to the ground is 300 m/s after being fired. What is the momentum and energy of the bullet? What is the momentum and energy of the rifle? What is the recoil speed of the rifle? Example: 1-D collision: 2 carts collide head-on on a frictionless track. The first cart has a mass of .5 kg and approaches the collision with a speed of 2 m/s. The second cart has a mass of .3 kg and approaches the collision with a speed of 2 m/s. After the collision, the second cart rebounds from the collision at a speed of 2 m/s. Determine the initial kinetic energies and final velocities and kinetic energies of both masses. How much energy is lost in this collision?

9. Example: 2-D collision. A 5.00 kg mass initially moves in the positive x-direction with a speed of 2.00 m/s, and then collides with a 3.00 kg mass which is initially at rest. After the collision, the first mass is found to be moving at 1.00 m/s 30º from the positive x-axis. What is the final velocity of the second mass? What is the total initial and final kinetic energy of the system?

10. Elastic and Inelastic Collisions • Elastic Collisions • interaction is conservative force • mechanical energy is conserved • no “stickiness” • Inelastic Collisions • interaction is not conservative force • some mechanical energy is lost • some “stickiness” • Completely Inelastic Collisions • interaction is not conservative force • maximum loss of mechanical energy • colliders stick together after the collision • In all collisions, momentum is conserved; in elastic collisions, energy is conserved as well.

11. Completely inelastic collisions • vA2 = vB2 = v2 so • mAvA1 + mBvB1 = (mA+ mB)v2 • take object B initially at rest (can consider as 1-d problem)  energy is always lost in a completely inelastic collision

12. Example: 1-D collision: 2 carts collide head-on on a frictionless track. The first cart has a mass of .5 kg and approaches the collision with a speed of 2 m/s. The second cart has a mass of .3 kg and approaches the collision with a speed of 2 m/s. After the collision, the the carts stick together. Determine the initial kinetic energies and final velocity and kinetic energy of both masses. How much energy is lost in this collision?

13. Example: Ballistic Pendulum. A bullet of mass m is fired into a block of wood of mass M, where it remains imbedded. The block is suspended like a pendulum, and swings up to a maximum height y. Relate M, m and y to the bullets initial velocity. Example: A 2000 kg car traveling east at 10 m/s collides (completely inelastically) with a 1000 kg car traveling north at 15 m/s. Find the velocity of the wreckage just after the collision, and the energy lost in the collision.

14. Elastic Collisions • examine 1-d elastic collision, with B at rest before collision

15. Example: 1-D collision: 2 carts collide elastically head-on on a frictionless track. The first cart has a mass of .5 kg and approaches the collision with a speed of 2 m/s. The second cart has a mass of . 3 kg and approaches the collision with a speed of 2 m/s. Determine the initial kinetic energies and final velocities and kinetic energies of both masses. Example: A neutron (mass 1 u = 1.66E-27 kg) traveling at 2.6E7 m/s strikes a carbon nucleus (mass 12 u). What are the velocities after the collision? by what factor is the neutron’s kinetic energy reduced by the collision?

16. Example: A spacecraft of mass 825 kg approaches Saturn “head on” with an initial speed of 9.6 km/s while Saturn (mass 5.69E26 kg) moves along its orbit at 10.4 km/s. The gravitational force of Saturn on the spacecraft swings the spacecraft back in the opposite direction. What is the final speed of the spacecraft. Example: An elastic (2-d) collision of two pucks on a frictionless air table occurs with the first mass ( 0.500 kg) approaching at 4.00 m/s in the positive x-direction and the second mass (0.300 kg) initially at rest. After the collision, the first puck moves off at a speed of 2.00 m/s in an unknown direction. What is the direction of the first pucks velocity after the collision, and what is the speed and direction of the second puck after the collision/

17. Center of Mass • aka Center of Inertia • “average” location of mass on a system of particles • motion of center of mass

18. External forces and the motion of center of mass example: A 50.0 kg woman walks from one end of 5m, 40.0 kg canoe to the other. Both the canoe and the woman are initially at rest. If the friction between the water and the canoe is negligible, how far does the woman move relative to shore? How far does the boat move relative to shore?