1. Chapter 7 Linear Momentum

2. Objectives: The student will be able to: Apply the laws of conservation of momentum and energy to problems involving collisions between two point masses.

3. Finish pHET in 2 dimensions

4. February 4, 2016 Two-Dimensional Collisions For a general collision of two objects in two- dimensional space, the conservation of momentum principle implies that the total momentum of the system in each direction is conserved

5. February 4, 2016 Two-Dimensional Collisions The momentum is conserved in all directions Use subscripts for –Identifying the object –Indicating initial or final values –The velocity components If the collision is elastic, use conservation of kinetic energy as a second equation –Remember, the simpler equation can only be used for one-dimensional situations

6. February 4, 2016 Glancing Collisions The “after” velocities have x and y components Momentum is conserved in the x direction and in the y direction Apply conservation of momentum separately to each direction

7. 7-7 Collisions in Two or Three Dimensions Conservation of energy and momentum can also be used to analyze collisions in two or three dimensions, but unless the situation is very simple, the math quickly becomes unwieldy. Here, a moving object collides with an object initially at rest. Knowing the masses and initial velocities is not enough; we need to know the angles as well in order to find the final velocities.

8. February 4, 2016 2-D Collision, example Particle 1 is moving at velocity and particle 2 is at rest In the x-direction, the initial momentum is m 1 v 1i In the y-direction, the initial momentum is 0

9. February 4, 2016 2-D Collision, example cont After the collision, the momentum in the x-direction is m 1 v 1f cos    m 2 v 2f cos  After the collision, the momentum in the y-direction is m 1 v 1f sin    m 2 v 2f sin  If the collision is elastic, apply the kinetic energy equation

10. February 4, 2016 Collision at an Intersection  A car with mass 1.5×10 3 kg traveling east at a speed of 25 m/s collides at an intersection with a 2.5×10 3 kg van traveling north at a speed of 20 m/s. Find the magnitude and direction of the velocity of the wreckage after the collision, assuming that the vehicles undergo a perfectly inelastic collision and assuming that friction between the vehicles and the road can be neglected.

11. February 4, 2016 Collision at an Intersection

12. February 4, 2016 Collision at an Intersection

13. Copyright © 2009 Pearson Education, Inc. 7-7 Collisions in Two or Three Dimensions Example : Billiard ball collision in 2-D. Billiard ball A moving with speed v A = 3.0 m/s in the + x direction strikes an equal-mass ball B initially at rest. The two balls are observed to move off at 45° to the x axis, ball A above the x axis and ball B below. That is, θ A ’ = 45° and θ B ’ = -45 °. What are the speeds of the two balls after the collision?

14. 7-7 Collisions in Two or Three Dimensions Problem solving: 1. Choose the system. If it is complex, subsystems may be chosen where one or more conservation laws apply. 2. Is there an external force? If so, is the collision time short enough that you can ignore it? 3. Draw diagrams of the initial and final situations, with momentum vectors labeled. 4. Choose a coordinate system.

15. 7-7 Collisions in Two or Three Dimensions 5. Apply momentum conservation; there will be one equation for each dimension. 6. If the collision is elastic, apply conservation of kinetic energy as well. 7. Solve. 8. Check units and magnitudes of result.

16. Homework Problems Chapter 7 40 and 42

17. Closure What did you learn about collisions in two dimensions?