Aim: How do we solve collisions in two dimensions?

1. Aim: How do we solve collisions in two dimensions?

2. Two Dimensional Inelastic Collision In a two dimensional collision, the momentum must be conserved in each dimension. In the x-direction, momentum is conserved. In the y-direction, momentum is conserved.

3. Conservation of Momentum in Two Dimensions Inelastic Collisions m1v1ix + m2v2ix = (m1 + m2)vfx m1v1iy + m2v2iy = (m1 + m2)vfy

4. Conservation of Momentum in Two Dimensions Elastic and Inelastic Collisions m1v1ix + m2v2ix = m1v1fx + m2v2fx m1v1iy + m2v2iy = m1v1fy +m2v2fy

5. Thought Question 1 Using conservation of momentum, sketch the path taken by the mass m1 after the collision. Explain in a sentence how you determined this. M1 will move in a northeast direction since momentum must be conserved in both the vertical and horizontal directions.

6. Thought Question 2 A 10 g mass moves to the right at 2 m/s. It explodes into three pieces m1 , m2 , and m3 . Without doing any calculations, predict the motion of m2 after the explosion. Justify your prediction by writing a sentence. M2 moves to the west because momentum must be conserved in both the horizontal and vertical directions.

7. Thought Question3 Consider the collision below between puck A and puck B. Puck has an initial momentum of 5 kg m/s and collides with puck B which is at rest. Arrows 1,2,and 3 show three possible paths taken by puck A after the collision. (see picture on next slide) Which path will puck A take if the momentum of puck B after the collision has an x component of (a) 5 kg m/s, (b) more than 5 kg m/s, and (c) less than 5 kg m/s? a) Path 2 b) Path 3 c) Path 1

8. Thought Question Diagram

9. Problem Two masses, m = 1 kg and M=5 kg are free to slide on a horizontal frictionless surface. The magnitudes and directions of the velocities are shown before and after the collision. • Write down the conservation of momentum equations for both the x and y directions. • Determine the speeds of the two objects after the collision. • Determine if this is an elastic collision.

10. Two Dimensional Momentum Problems

12. Sample Problems 1. A 1325 kg car, C, moving north at 27.0 m/s, collides with a 2165 kg car, D moving east at 11.0 m/s. The two cars are stuck together. In what direction and with what speed do they move after the collision?

14. Sample Problems 2. A 1345 kg car moving east at 15.7 m/s is struck by a 1923 kg car moving north. They are stuck together and move with an initial velocity of 14.5 m/s at Θ=63.5◦. Was the north-moving car exceeding the 20.1 m/s speed limit?

16. Elastic Collisions Remember elastic collisions conserve kinetic energy, but kinetic energy is a scalar quantity so we just write m1(v1i)2 + m2(v2i)2 =m1(v1f)2 +m2(v2f)2

17. 3. A proton collides elastically with another proton that is initially at rest. The incoming proton has an initial speed of 3.5 x 105 m/s and makes a collision with the second proton. After the collision, one proton moves off at an angle of 37 degrees to the original direction of motion and the second deflects at an angle of ф to the same axis. Find the final speeds of the two protons and the angle ф.

20. Sample Problem 4. A billiard ball of mass = 0.400 kg moving with a horizontal velocity of 2 m/s strikes a second ball, initially at rest, of the same mass. As a result of the collision, the first ball is deflected off at an angle of 30◦ with a speed of 1.20 m/s. • Write down the equations for the components of momentum in the x and y-direction separately. • Solve these equations for speed of the second ball and angle Θ. State whether the collision is elastic or inelastic.