Fall 2013 Physics 172 – Recitation 2 Updating the Momentum Solution Purpose: The purpose of this recitation is to give you experience working with momentum and the momentum update formula. Readings: Chapter 2.1-2.4 Learning Objectives: 2.2.1 Write down the momentum principle, including subscripts 2.2.2 Explain what is meant by "net" force 2.3.1 Apply the momentum principle (in update form) to solve problems involving the motion of objects 2.3.2 Calculate the approximate average velocity of an object and describe when it is exactly correct 2.4.1 Use the momentum update formula to relate changes in the momentum of an object (or a system) to the (possibly changing) net external forces during the time interval. Today you will work in a group of 4 to determine a solution to today’s challenge problem. While working in a group: • All members of the group should participate in determining the solution. • Make sure that all members discuss and agree on the answer to each part of the problem. • Make sure your group’s solution is neatly and clearly written on your dry erase board. • Be prepared to present your work to the rest of the class. • Be prepared to answer questions posed by other students or your TA. • Be prepared to justify any answers you provide by giving physical reasons.

Challenge Problem: The figure below shows positions of a 0.16 kg hockey puck on the horizontal ice surface of a hockey rink at a sequence of times t1 , t2 , t3 , t4 and t5 that are spaced 0.1 seconds apart. You are looking down onto the surface of the rink.

y 1 2 3

4

5

x

The x-axis of the coordinate system shown runs along the length of the rink, its y-axis runs across the rink and its z-axis points up toward you. The spacing between the gridlines shown is 2 meters. At certain time a hockey player (not shown) hits the puck with his hockey stick. • Based on the information given above and in the graph, determine the components of the Force that the hockey player hits the puck with. Assume that the hockey duration of the hit of the hockey puck is very short t 0.005. Δ c = • Be sure to state any reasonable approximations or assumptions that you made to solve this problem.

You will need to draw the diagram and show the steps of GOAL1 on your white board. If you know how to do the problem, you need to show the steps in your solution.

Gather Information: The information given in the problem is the mass of the hockey puck and the position of the puck at several different types. We can fill in the following table:

Mass of the Hockey Puck: 0.16 kg

Time (seconds) Position

t1 0 < 0 , 6 , 0> m

t2 0.1 < 2 , 6 , 0> m

t3 0.2 < 4 , 6 , 0> m t 0.3 < 5 , 4 , 0> m € 4 t 0.4 < 6 , 2 , 0> m € 5

The goal€ of this problem is to determine the Force striking the hockey puck. € kgm The units€ of Force are Newtons, N, or 2 . A force is a vector, which will have magnitude and s direction.

Outline Your Approach When making a plan €it may be appropriate to work backwards from what we want until we get what we are given. • We want the force that the hockey pucker hits the puck with  • We can get the net force from the momentum update formula, where p is the momentum  i before the puck is hit and p f is right after the puck is hit.    p − p = F (t − t ) f i net f i rearranging we have:   €  p p f − i € Fnet = € Δt Note that the denominator is the duration of the collision between the hockey stick and the puck.

1 GOAL = Gather, Outline, Analyze, Learn Because the puck is on a low friction surface, ice, we can assume that the frictional force is negligible. In addition, if we assume the air resistance is negligible, we can   approximate F stick ≈ Fnet • We can determine the momentum from its definition:   1 p ≡ γmv , where γ =  2 € $ v ' 1 −& ) c % ( • To determine the momentum we need the instantaneous velocity, which is given by: €      dr dx dy dz v = = , , . dt dt dt dt € Since we cannot determine the instantaneous velocity directly, we will have to approximate   the instantaneous velocity, v ≈ v , as the average velocity. avg • We can determine the average velocity from the change in position over time. €     Δrfi rf − ri v = = fi Δt t − t fi f €i • The only thing left to determine is the time at which the player hits the puck occurs. So our plan for determining the solution, is simply these steps in reverse: 1. Determine the time at which the player hits the puck. € 2. Determine the average velocity before and after the player hits the puck. 3. Approximate the momentum before and after the player hits the puck. 4. Use the momentum principle to determine the net force and thus the approximate force that the player hit the puck.

Analyze the Problem:

Step 1: Determine the time the player hits the puck.

To determine the time the player hits the puck, we have to look for evidence of an interaction.

At t3 we observe that the puck changed direction and its speed has increased, both of which are

evidence of an interaction. We can conclude that player hit the puck at t3.

Step 2: Determine the Average Velocity Before and After t . € 3

We need to determine two average velocities, that between t € and t and that between t and t .    2 3 4 3  Δr r − r 4,6,0 m − 2,6,0 m 2,0,0 m v 32 3 2 20,0,0 m 32 = = = = € = s Δt32 t3 − t2 0.2s − 0.1s 0.1s and    € € € €  Δr r − r 5,4,0 m − 4,6,0 m 1,−2,0 m v = 43 = 4 3 = = = 10,−20,0 m 43 Δt t − t 0.3s − 0.2s 0.1s s € 43 4 3

Step 3: Approximate the Momentum Before and After t3.

€

€   We know that the momentum is determined by the relationship, p ≡ γmv . We can then determine   and . p 32 p 43   kgm p = γmv = γ0.16kg 20,0,0 m = γ 3.2,0,0 32 32 s s We still need to calculate gamma. Plugging in our known€ information we see that 20m/s is much € €smaller than the speed of light, so gamma =1 1 1 1 γ = = ≈ = 1 €  2 2 1 $ v ' $ 20m ' 1 −& ) 1 & s ) c − 8 % ( & 3.0x10 m ) % s (   kgm p = mv = 0.16kg 20,0,0 m = 3.2,0,0 32 32 s s

 We can calculate, , using the same method. € p 43   kgm p = mv = 0.16kg 10,−20,0 m = 1.6,−3.2,0 . € 43 43 s s

Step 4: Determine€ the Approximate Force that the Player Hits the Puck with

€ We can now use the momentum update formula to determine the net force.

  kgm kgm kgm  p p   1.6,−3.2,0 − 3.2,0,0 −1.6,−3.2,0 f − i p43 − p32 s s s kgm Fnet = = = = = −320,−640,0 2 Δt 0.005s 0.005s 0.005s s

This answer makes sense, because the force has components in the –x direction and the –y direction, which is how the motion of the puck changes. In addition, the strength of the force is within the range that we would expect to cause the puck to change its direction and speed.

Learn From Your Efforts: From this experience, you have hopefully learned the following:    • How to use the momentum update formula to determine the Net Force: p − p = F Δt f i net • How you can use position and time data to determine when an interaction occurs • How you can use the position and time data to determine the strength of the force causing the change in motion of an object.