Linear Momentum and Collisions

1. Linear Momentum and Collisions Mr. Kamanda

2. Objectives By the end of this unit, you should be able to: • Define linear momentum as product of mass and velocity, and calculate momentum of a body. • Define Impulse as change in momentum or the product of force and time, and calculate the impulse on a body. • Write out an expression/ equation to represent the principle of conservation of linear momentum. • Apply the principle of conservation of linear momentum to solve problems. • Distinguish between elastic and inelastic collisions.

3. Momentum • The linear momentum of an object of mass m moving with a velocity v is defined as the product of the mass and the velocity • p = m v • SI Units are kg m / s • Vector quantity, the direction of the momentum is the same as the velocity’s

4. Questions. • What is the momentum of a 70 kg runner travelling at 10 m/s? • What is the momentum of a 800kg car travelling at 20 m/s?

5. Questions: • What is the change in momentum of a 950kg car that travels from 40 m/s to 31 m/s. • A 0.095 kg tennis ball travelling at 40 m/s bounces off a wall with a speed of 30m/s. What is the change in momentum of the ball?

6. Impulse • In order to change the momentum of an object, a force must be applied • Recall Newton’s second law:

7. Impulse cont. • FΔt is defined as the impulse, J. • Vector quantity, the direction is the same as the direction of the force • SI unit of Impulse: Kgm/s or Ns • If the force is not constant, use the average forceapplied

8. Questions • A 0.35 kg volleyball is spiked so that its incoming velocity of 4 m/s is changed to an outgoing velocity of -21m/s. What impulse does the player apply to the ball? • A golfer, driving a golf ball off the tee, gives the ball a velocity of 38 m/s. The mass of the ball is 0.045 kg, and the duration of the impact with the golf ball is 3.0 x 10-3 s. (a) What is the change in momentum of the ball? (b) Determine the average force applied to the ball by the club?

9. Impulse from a force-time graph • The impulse imparted by a force during the time interval Δt is equal to the area under the force-time graph from the beginning to the end of the time interval

10. Conservation of Momentum • Momentum in an isolated system in which a collision occurs is conserved • The principle of conservation of momentum states when no external forces act on a system consisting of two objects that collide with each other, the total momentum of the system before the collision is equal to the total momentum of the system after the collision

11. Conservation of Momentum, cont. • Mathematically:

12. Questions: • An 85.0 kg fisherman jumps from a dock into a 135.0 kg rowboat at rest on the west side of the dock. If the velocity of the fisherman is 4.30 m/s to the west as he leaves the dock, what is the final velocity of the fisherman and the boat? • A rifle with a weight of 30 N fires a 5.0-g bullet with a speed of 300 m/s. (a) Find the recoil speed of the rifle. (b) If a 700-N man holds the rifle firmly against his shoulder, find the recoil speed of man and rifle.

13. Types of Collisions • Momentum is conserved in any collision • Inelastic collisions • Kinetic energy is not conserved • Perfectly inelastic collisions occur when the objects stick together • Elastic collision • both momentum and kinetic energy are conserved • These are very rare and usually microscopic. Collision between two hard steel balls is a close approximation.

14. More About Perfectly Inelastic Collisions • When two objects stick together after the collision, they have undergone a perfectly inelastic collision • Conservation of momentum becomes

15. More About Elastic Collisions • Both momentum and kinetic energy are conserved

16. Problem Solving for One-Dimensional Collisions • Write the expressions for the momentum of each object before and after the collision • Remember to include the appropriate signs • Write an expression for the total momentum before and after the collision • Remember the momentum of the system is what is conserved

17. Problem Solving for One-Dimensional Collisions • If the collision is inelastic, solve the momentum equation for the unknown • Remember, KE is not conserved • If the collision is elastic, you can use the KE equation to solve for two unknowns

18. Questions • A 7.00-kg bowling ball collides head-on with a 2.00-kg bowling pin, which was originally at rest. The pin flies forward with a speed of 3.00 m/s. If the ball continues forward with a speed of 1.80 m/s, what was the initial speed of the ball? Ignore rotation of the ball.

19. Questions cont’d 2. Two 0.40 kg soccer balls collide elastically in a head-on collision. The first ball starts at rest, and the second has a speed of 3.5 m/s. After the collision, the second ball is at rest. a. What is the final speed of the first ball? b. What is the kinetic energy of the first ball after the collision? c. What is the kinetic energy of the second ball after the collision?

20. Questions cont’d 4. A 25.0 kg bumper car moving to the right at 5.00 m/s overtakes and collides elastically with a 35.0 kg bumper car moving to the right. After the collision, the 25 kg bumper car slows to 1.50 m/s to the right, and the 35.0 kg car moves at 4.50 m/s to the right. a. Find the velocity of the 35 kg bumper car before the collision. b. Verify your answer by calculating the total kinetic energy before and after the collision.

21. Summary • Linear Momentum, P = mv • Impulse, J = Ft =  P • Linear Momentum and impulse are vector quantities. • Linear Momentum is always conserved in a collision. • Kinetic energy is only conserved in elastic collision.