Momentum and Impulse

1. Momentum and Impulse

2. Linear Momentum • Momentum is defined as mass times velocity. • Momentum is represented by the symbol p, and is a vector quantity. p = mv momentum = mass  velocity • Momentum has the dimensions of mass x length/time (kg x m/s)

3. Example • A 2250 kg pickup truck has a velocity of 25 m/s to the east. What is the momentum of the truck? p = mv (2250 kg)(25 m/s east) 5.6 x 104 kg x m/s to the east

4. Your Turn • A deer with a mass of 146 kg is running head-on toward you at a speed of 17 m/s. You are going north find the momentum of the deer. p = mv (146 kg)17 m/s north) 2482 kg x m/s to the north

5. Linear Momentum • Impulse • The product of the force and the time over which the force acts on an object is called impulse. • The impulse-momentum theorem states that when a net force is applied to an object over a certain time interval, the force will cause a change in the object’s momentum. F∆t = ∆p = mvf – mvi force  time interval = change in momentum Force is reduced when the time interval of an impact is increased.

6. Example • A 1400 kg car moving westward with a velocity of 15 m/s collides with a utility pole and is brought to rest in 0.30 s. Find the force exerted on the car during the collision. F∆t = ∆p = mvf – mvi F = mvf – mv ∆t F = (1400kg)(0 m/s) – (1400 kg)(-15 m/s) 0.30 s F = 7.0 x 104 N to the east

7. Conservation of Momentum

8. Momentum is Conserved • So far we only have considered the momentum of only one object at a time. • Now we will look at two or more objects interacting with each other. • Picture this. . . • You are playing pool. You strike the cue ball it hits the 8 ball. The 8 ball had no momentum before they collided. • During the collision the cue ball loses momentum and the 8 ball gains momentum. • The momentum the cue ball loses is the same amount that the 8 ball gained.

9. Momentum is Conserved • The Law of Conservation of Momentum: The total momentum of all objects interacting with one another remains constant regardless of the nature of the forces between the objects. m1v1,i+ m2v2,i = m1v1,f + m2v2,f total initial momentum = total final momentum

10. Momentum is Conserved • Picture this . . . • Two people on skates facing one another. They push away from one another. Initially, they are both at rest with a momentum of 0. When the push away, they move in opposite directions with equal but opposite momentum, so that the total momentum is 0.

11. Example • A 76 kg boater, initially at rest in a stationary 45 kg boat, steps out of the boat and onto the dock. If the boater moves out of the boat with a velocity of 2.5 m/s to the right, what is the final velocity of the boat?

12. Solution • m1 = 76 kg • m2 = 45 kg • v1,i = 0 • v2,i = 0 • v1,f = 2.5 m/s Right • v2,f = ? • m1v1,i + m2v2,i = m1v1,f +m2v2,f • Because the boater and the boat are initially at rest, the total initial momentum of the system is equal to zero. Therefore: • m1v1,f + m2v2,f = 0 Rearrange the equation to solve for the final velocity of the boat. v2,f= - m1v1,f/m2 v2,f= - (76 kg) (2.5 m/s)/(45 kg) v2,f = -4.2 m/s The negative sign for v2,f indicates that the boat is moving to the left, in the direction opposite the motion of the boater.

13. Elastic and Inelastic Collisions

14. Types of Collisions

15. Collisions • Perfectly inelastic collision • A collision in which two objects stick together after colliding and move together as one mass is called a perfectly inelastic collision. • Example: The collision between two football players during a tackle. • Conservation of momentum for a perfectly inelastic collision: m1v1,i + m2v2,i = (m1 + m2)vf total initial momentum = total final momentum

16. Example • A 1850 kg luxury sedan stopped at a traffic light is struck from behind by a compact car with a mass of 975 kg. The two cars become entangled as a result of the collision. If the compact car was moving with a velocity of 22.0 m/s to the north before the collision, what is the velocity of the entangled mass after the collision? m1= 1850 kg m2= 975 kg v1,i= 0 m/s v2,i= 22.0 m/s north vf = ? m1v1,i + m2v2,i = (m1 + m2)vf vf=m1v1,i + m2v2,i/ (m1 + m2) vf=(1850 kg)(0 m/s) + (975 kg)(22.0 m/s) (1850 kg + 975 kg) vf=7.59 m/s north

17. Kinetic Energy in Inelastic Collisions • In an inelastic collision the total kinetic energy does not remain constant when the objects collide and stick together. • Some energy is converted into sound energy and internal energy as the objects deform during the collision. • Elastic in physics refers to a material that when work is done to deform the material during a collision the same amount of work is done to return the material to its original shape. • Inelastic material does not return to its original shape and therefore some energy is converted to sound or heat.

18. Elastic Collisions • A collision in which the total momentum and the total kinetic energy are conserved is called an elastic collision. • Elastic means that after a collision the objects remain separated. • Two objects collide and return to their original shapes with no loss of total kinetic energy. After the collision the two objects move separately. • Both the total momentum and total kinetic energy are conserved.

19. Real Collisions • Most collisions are not perfectly inelastic (they don’t stick together and move as one) • Most collisions are not elastic. • Even nearly elastic collisions result in some decrease of kinetic energy. • A football deforms when kicked • A sound is produced (sound signifies a decrease in kinetic energy)

20. Elastic Collisions • The total momentum is always constant throughout the collision. In addition, if the collision is perfectly elastic, the value of the total kinetic energy after the collision is equal to the value before the collision.

21. Impulse-Momentum Theorem • Consider the egg. On an index card explain what is happening. Relate this example to the purpose of an air bag in a vehicle.