Contents. Objectives Velocity Addition CM Velocity 2 D Collisions totally inelastic elastic Recap. Contents

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Physics 121 for Majors totally in Class 16 totally in Velocity Addition

Energy Conservation of Energy Power totally in Today s Class Adding

energy to problems in 2 and 3 dimensions totally in Section 1 Velocity

1 Physics 121 for Majors totally in Class 16 totally in Velocity Addition and Collisions In Two Dimensions Last Class More on Work Potential Energy Conservation of Energy Power totally in Today s Class Adding velocities in 2 and 3 dimensions Applying conservation of momentum and energy to problems in 2 and 3 dimensions totally in Section 1 Velocity Addition totally in Rowing Across a River If you are in a still lake, you can comfortably row your boat at a speed of 1.00 m/s. You wish to cross a river that is 200 m across and flows at a rate of 2.00 m/s. totally in 1

2 Some Questions If you are in a still lake, you can comfortably row your boat at a speed of 1.00 m/s. You wish to cross a river that is 200 m across and flows at a rate of 2.00 m/s. How far do you travel with respect to the water? How long does it take to get across the river? How far downstream will you go? How far will you travel with respect to the shore? totally in One Last Question What would you have to do get straight across the river? totally in Airplane --- How would you land in a crosswind? totally in Landing in a Crosswind totally in The Math totally in Section 2 totally in We just need to add the velocities as vectors. Center of Mass Velocity For example: ab is boat to river bc is river to bank ac is boat to bank 2

3 Center of Mass Two objects: Where is the center of mass located? totally in Center of Mass The velocity of the center of mass is totally in Two hockey pucks, each with mass = 160 g, have velocities What is? totally in What are the velocities in the cm frame? What is the total momentum in the cm? Would that still be true if the masses were different? totally in Section 3 2-D Collisions totally in A Complication -- When two hockey pucks collide, they can rotate. For now we ll assume that the collisions have no rotation. If in, the collisions will be head-on in the cm frame. totally in If, the colliding surfaces are frictionless. 3

4 totally in collisions totally in Totally In Collision One hockey puck collides with a second hockey puck that has a third hockey puck stacked on top. The velocities before the collision are: What is the velocity after the collision? Show that the total momentum is zero in the cm frame. totally in Totally In Collision Conclusion: Find the velocity of the cm. This is velocity of the combined mass after the collision since the total momentum is zero before and after the collision in the cm totally in collisions totally in One method is conservation of momentum and energy. totally in totally in There are three equations but four unknowns Why? 4

Let s assume we know the direction of the smaller mass: cos, sin

quite hard to solve without Mathematica, and very hard to solve if you

Usually an easier method is to use the cm frame.

00 2.00 2 totally in Find the cm velocity and the velocities in the cm

2 1 4 2 2 1 4 2 2 1 The collision will change the directions of both

5 Let s assume we know the direction of the smaller mass: cos, sin totally in Mathematica with 10 totally in cos 2 6 sin This is quite hard to solve without Mathematica, and very hard to solve if you don t put in numbers for the known variables. Usually an easier method is to use the cm frame. We ll do the hardest version of the problem totally in totally in Find the cm velocity and the velocities in the cm frame The collision will change the directions of both objects, but not the speeds If mass 1 comes off at an angle, mass 2 comes off in the opposite direction. totally in cos 20 sin 5cos 5sin Let s go back to the lab frame totally in 5

20 cos 20 sin 5cos 5sin 20 cos 2 20 sin 1 5cos2 5sin1 But we probably know the direction in the lab

totally in 20 cos 2 20 sin 1 5cos2 5sin1 We do know : 20 sin 1 tan 20 cos 2 And this is a little tricky,

totally in 202 tan1 20 1 20 4 20 4 tan 20 20x Now we have a quadratic equation in cos.

6 20 cos 20 sin 5cos 5sin 20 cos 2 20 sin 1 5cos2 5sin1 But we probably know the direction in the lab frame, not the cm frame! totally in 20 cos 2 20 sin 1 5cos2 5sin1 We do know : 20 sin 1 tan 20 cos 2 And this is a little tricky, but not too hard, without Mathematica totally in 20 sin 1 tan 20 cos 2 20 cos 2 tan 20 sin 1 totally in totally in 202 tan tan 20 20x Now we have a quadratic equation in cos. Conclusion: Find the velocity of the cm. Subtract off each velocity to get cm velocities Change directions Add cm velocity to each velocity to get final lab velocities totally in Section 4 Recap totally in 6

7 Big Ideas Adding velocities as vectors is useful for airplanes and boats Collisions in 2-D and 3-D are like 1-D, except we have to deal with vectors It is usually easier to go to the cm frame, collide, and then go back to the lab Remember that we assume no rotational motion! totally in Schedule Do Post-Class Quiz #16 Do Pre-Class Quiz #17 HW #16 is due Wednesday Quiz #5 is due Nov. 11 totally in 7

Contents. Contents. Contents

Contents. Contents. Contents Physics 121 for Majors Schedule HW #5 is due Friday Quiz #2 is due 9/29 Lab #2 is due Monday Midterm 1 is 10/2 in the classroom. Class 6 and Collisions Relative Velocity Last Class Natural motion is straight-line