During an ice show a 60 kg skater

PHYS 183 Paper and Pencil Homework Guidelines grading

Paper and pencil homework assignments are worth 50 points each (somewhat more than a typical MP assignment), with individual problems weighted according to difficulty and expected effort. Grades will be based on thoroughness and clarity, as well as correctness. Partial credit may be given for incomplete or incorrect solutions, as long as some appropriate concepts and techniques are applied. Credit can only be given, however, if it is clear to the instructor/grader that such is the case. See below for guidelines on homework format.

Solutions to each HW problem will be posted on Canvas on the day that the assignment is due. Therefore, no late assignments can be accepted. It will be your responsibility to review the posted solutions and to identify specific errors (if any) in the work you submitted. Some problems will be graded by me and some will be graded by a Teaching Assistant. Comments by the grader on your homework may or may not contain details about where you may have made mistakes. If, after reviewing the solutions, you still have questions about how to solve a problem, please visit me during office hours or make an appointment.

format To achieve a high homework grade, you must demonstrate clearly that you understand the process by which you arrive at a solution. Merely obtaining a correct numerical answer is insufficient. Unless explicitly instructed otherwise, you are expected to provide the following steps for each problem (or separate part of each problem):

1. Draw a sketch that illustrates the problem and shows the coordinates. If a sketch is provided in the problem statement, you can reproduce that. On or with the sketch you should...

a. describe the coordinate system you will use. For example, which way is +x? b. identify what constitutes the system to which you are applying the equation(s), if appropriate c. describe each symbol that you plan to use in your solution. Label important features of your

sketch with these symbols. 2. Describe any simplifications or assumptions you can make as you set up the problem. For example, can

you model something as a "point charge" or an "infinite line of charge"? 3. State the basic physics principle(s) (e.g. Conservation of Energy, or Coulomb's Law) that you are using to

solve the problem and write the generic form of the equation(s) that express the principle(s). 4. Using any assumptions or simplifications from part 1, write your equations or formulas as they apply to

this particular problem. You are still using only symbols at this point, all of which (except for constants) should be identified in step 1c.

5. Identify the quantity (using your symbol) that you need to solve for. 6. Do the necessary algebra to solve for this unknown - still using only symbols.

** The process of steps 3-6 may involve connecting more than one equation that involves different physical principles. The need to do this makes a problem more than just a "plug and chug" exercise. Explaining how you make such connections is the heart of each pencil & paper homework. In order to convince me that you can do this (rather than just copying or looking up an answer), you have to explain your reasoning.

7. If numbers are given, plug them in and calculate a final numerical answer.

A model solution illustrating these steps is provided below. Expect about one full page per problem.

Collaboration on homework problems is allowed – even encouraged. However, this is not an excuse for plagiarizing homework! Even if someone helped you figure out a problem, you should be able to explain it in your own words. This is why I will be grading thoroughness. If you copy someone else’s thorough solution, it will be obvious and I will mark both papers down severely. If the answer is not thorough, I will assume you didn’t thoroughly understand it.

While you are working a given problem, possibly collaborating with classmates, make a draft solution. It can be messy and full of mistakes. This is the purpose of making a draft - this is NOT what you submit for credit. I will hold you to a very high standard of neatness! If I have difficulty reading your homework, I will give it back to you to redo. Clearly, a satisfactory homework cannot be prepared at the last minute. Start early!

Model Solution (problem #37 in chapter 9 of Knight)

Problem statement

At the center of a 50 m diameter circular ice rink, a 75 kg skater traveling north at 2.5 m/s collides with, and holds onto, a 60 kg skater who had been travelling west at 3.5 m/s. a) How long will it take them to glide to the edge of the ice rink? b) Where will they reach the edge? Give your answer as an angle north of west.

Solution to part (a)

step 1: I can calculate the time it takes for the combined skaters to travel from the center to the edge by using kinematics: Δt = Δx/v , where v is their velocity after collision. To obtain that, I will apply conservation of momentum: 𝑝! = 𝑝! . Since this is a two-dimensional problem, the conservation of momemtum applies separately to each component, so there are two equations: 𝑝!,! = 𝑝!,! and 𝑝!,! = 𝑝!,!

step 2: The system consists only of the two skaters. If the ice rink is frictionless and horizontal, and if we neglect air resistance, then there is no net force acting on this system. The coordinate system will have +x pointing east, and +y pointing north. It is at rest with the ice rink.

step 3: Let m1 and 𝑣! refer to the mass and velocity, respectively, of skater 1. Likewise, m2 and 𝑣! will represent the mass and velocity of skater 2. Subscripts i and f will indicate initial and final values.

step 4: In this problem, skater 1 is moving north and skater 2 is moving west, so the initial momentum is just 𝑚!𝑣!! in the x-direction and 𝑚!𝑣!! in the y-direction. Note that 𝑣!! is negative (since east is positive). Since this is an elastic collision, the final state of the system consists of the combined masses of the skaters moving with a single velocity, which has two components and 𝑣!" and 𝑣!".

So, the conservation of momentum in the x-direction can be written: 𝑚! 𝑣!! = (𝑚! +𝑚!) 𝑣!" and in the y- direction the equation is: 𝑚! 𝑣!! = (𝑚! +𝑚!) 𝑣!"

step 6: We want to solve for the two components of the final velocity, 𝑣!" and 𝑣!".

𝑣!" = 𝑚! 𝑣!! 𝑚! +𝑚!

= (60 kg)(−3.5 m/s) (75 kg + 60 kg)

= −210 kg∙m/s

135 kg = −1.56 m/s

𝑣!" = 𝑚! 𝑣!! 𝑚! +𝑚!

= (75 kg)(2.5 m/s) (75 kg + 60 kg)

= 187.5 kg∙m/s

135 kg = +1.39 m/s

Now that we have the components, we can calculate the final speed:

𝑣! = (𝑣!")! + (𝑣!")! = (−1.56 m/s)! + (1.39 m/s)! = 2.09 m/s

Now that we have the speed, we can calculate how long it takes to reach the edge:

Δ𝑡 = !! !! = !" m

!.!" m/s = 23.9 s

Solution to part (b) The question asks for the angle (north of west) of their final motion. This is determined by the components of their final velocity, except that we need to reverse the sign of the x-component (so that west is positive). As shown in the figure, the angle is given by θ = tan!! !!"

!!" = tan!! !.!"

!.!" = 41.7°.