Some Thoughts on Mathematical Solutions of Physics Problems in AP Physics 1 1. There has been a continuing discussion on the AP Community Bulletin Board concerning the role and importance of the ability to carry out mathematical solutions to physics problems in the AP Physics 1 curriculum and their value in preparing for the AP Physics 1 exam. 2. The discourse has occasionally used the somewhat pejorative slang term of “plug and chug” to describe this kind of problem solving, which can be interpreted as down-playing the ability to carry out mathematical solutions to that sort of problem. My first suggestion is to drop that usage from our discussion. That kind of problem solving is more properly referred to as a form of means-ends problem solving as discussed by George Polya in his well-known book from 1945, How to Solve It. For a brief summary see https://math.berkeley.edu/~gmelvin/polya.pdf (referenced 9/14/2015). I recommend the book. 3. Briefly (see the link for more detail) Polya has four principles: understand the problem, devise a plan, carry out the plan, and look back. For a purely algebraic solution, this might be carried out by a) listing the known quantities and those to be calculated, b) identifying and writing down the physically relevant equations, c) combining the equations and solving for the unknowns, d) substituting numerical values, calculating a result, and checking that units are correct and that numerical results are physically reasonable. 4. The procedure described is reasonably consistent with what high school students bring with them as their concept of the objective of problem solving: get a numerical result. This is a necessary, and important skill for physics students, and for students of any field that involves quantitative analysis, and should be part of the skill set developed in the AP Physics 1 course. 5. The skill described is necessary, important, but not the only skill needed for understanding and thinking in physics. In paragraph 3, step c notice that only after the equations have been solved are numerical values and units substituted and a numerical calculation made - not the usual practice of students in math classes, and not the typical procedure required by rubrics for exam problems in AP Physics B. However, the algebraic solution carries additional information and allows additional reasoning, if students properly understand mathematics. The algebraic expression constitutes a mathematical model of a general physical situation for which the numerical values are simply one instance. The algebraic expression allows students to reason using ratios – if quantity a is doubled or tripled, what happens to quantity b, for example. The expression can be solved for a different quantity, when a different set of knowns and unknowns are specified. This mathematical analysis represents the solution to a whole set of “problems,” and with practice can be recognized as such by students.

6. Furthermore, such a solution, and the process of analyzing a situation, can be re-represented in different forms, such as graphs, diagrams and even written paragraphs, since the mathematics in physics is not just a computational tool, but an expression of relationships that have been derived and idealized from experience and experiment. The mathematics is just one possible expression of a conceptual framework of physics. We would like students to build a set of intertwined, rich representations of what we know about the physical world. 7. This note is already too long. My point is that, yes, students need mathematical problem solving skills, but students also need the ability to work with the concepts of physics in other forms. I will post on Pretty Good Physics a copy of this note, supplemented by a description of where I see the mathematics in problem 5 of the 20015 AP Physics 2 exam and also an example of a paragraph length response problem that I wrote showing how mathematical reasoning would be used to arrive at an acceptable paragraph length response.

Where’s the math? 2015 AP P1 Q5 (7 points 13 min) Problem statement The figure above shows a string with one end attached to an oscillator and the other end attached to a block. The string passes over a massless pulley that turns with negligible friction. Four such strings, A, B, C, and D, are set up side by side, as shown in the diagram below. Each oscillator is adjusted to vibrate the string at its fundamental frequency f. The distance between each oscillator and pulley L is the same, and the mass M of each block is the same. However, the fundamental frequency of each string is different.

Parsing the problem statement Four strings

1. Same suspended mass M 2. Same length L 3. Each oscillating at fundamental frequency 4. Fundamental frequencies are different

From information given, expect student to be able to:    

a. Conclude M at rest, so tension force = Mg same for all strings

b. Standing wave at fundamental looks like with wavelength 2L c. Recognize wavelength is same for all strings

d. Wave equation is so same wavelength and different frequency means different wave velocities.

FT Fg

Parsing information and part a) The equation for the velocity v of a wave on a string is

where FT is the tension of the string and m/L is the mass per unit length (linear mass density) of the string. (a) What is different about the four strings shown above that would result in their having different fundamental frequencies? Explain how you arrived at your answer. The two velocity equations can be set equal (MATH) to get:    Which can be solved for frequency in terms of givens: From givens and reasoning above, only quantity in expression on right that is not known to be the same for all strings is m, the mass of the string. Thus linear mass densities (m/L) must be different to give different frequencies. Good Student Answer for part a)

For all four strings: 1. M is same so tensions are same. 2. L is same so fundamental wavelength is same, so from a different frequency requires a different velocity

3. Since velocity depends on tension and linear mass density, from equation given for velocity, with same tension must have different linear mass density.

Students could write more of their analysis, but above statements or equivalent would garner full credit (3 points) for part a). Common errors 1. Not mentioning wavelength and length relation. 2. Confusing M and m and saying tension must be different. 3. Stating that velocity must be same since they are all strings so wavelengths must be different. 5. Not relating wave equation velocity to velocity from tension and mass density. 5. Citing experience with guitar tuning to say tensions must be different.

Problem Statement part b)

(b) A student graphs frequency as a function of the inverse of the linear mass density. Will the graph be linear? Explain how you arrived at your answer.

Referring to equation assembled in part a) or assembling it here. Student should arrive at some form of relation at right. (MATH) (1 point) Student explanation: Putting two equations together to get has reciprocal of mass density inside square root sign, so to get linear graph would need to plot f vs. square root of (1/ mass density). (1 points) Common errors – 1. Failing to either refer to equation from a) or put equation together in part b). 2. Stating that graph would not be linear but would be “exponential.” (???Where are students getting term “exponential” instead of quadratic or power law???) Problem Statement part (c) (c) The frequency of the oscillator connected to string D is changed so that the string vibrates in its second harmonic. On the side view of string D below, mark and label the points on the string that have the greatest average vertical speed. Student could recognize second harmonic has 2 nodes and 3 antinodes, sketch it correctly, & mark antinodes as points of greatest average speed or could calculate as shown below.

Greatest Average Speeds

Mathematical analysis. At fundamental the frequency of string D was given as 350 Hz. At second harmonic frequency is given as 700 Hz.

Since this is the same string and under the same tension in both cases then

Common Errors Some students drew answer at right which got 1 point if antinodes marked for greatest speed (Probably confusing second harmonic with second overtone). A few stated it was second overtone or referred to musical knowledge implying that idea, in a few cases sketching waveforms for fundamental, first overtone and second overtone. A small percentage drew diagram like that at right with antinode at oscillator, apparently conflating standing waves on string with those in a tube open at one end. 0 points A significant number sketched waves like that at right, using heights of dashed lines given for linear measurement as amplitude indicators, with variety of wavelengths – usually four

or two units. 0 points

!1 f1 = !2 f2

!2 =f1f2!1

!2 =350hz700hz

(2L) = L

How  to  approach  a  paragraph  response  –  Suggestions  for  students  Robert A. Morse General  Suggestions  

• Read the whole problem • Mark off scratch area and outline solution as series of short

statements/equations/sketch graphs • Number them in a logical sequence – start, steps, conclusion • Rewrite in answer space in form of short sentences with equations or sketch

graphs as needed – as if you were explaining to another student • Keep it short and to the point.

See Also : Val E Monticue posted the following on 06/21/2015 14:00 EDT https://apcommunity.collegeboard.org/group/apphysics/discussion-boards/-/message_boards/view_message/65042331 RE: What I learned as a 4th year Reader (Acorn Table Leader) 2015 An  example  of  a  possible  paragraph  length  response  question  –  part  1  

A heavily loaded barge is connected to a tugboat by a cable. The barge and tug are initially at rest. At t = 0, the tug begins pulling on the cable with a constant tension. The table below shows the speed of the barge during the first 15 minutes of its motion.

Time (min) Speed (km/h) student work: 0 0 increase in speed 5 4 +4 10 6 +2 15 7 +1

a) Based on the description and data, assuming there are no changes in the conditions, what do you expect will happen to the speed of the barge during the next 15 minutes of motion? Briefly explain your reasoning.  (A paragraph length response question (7 points, 12 minutes) would usually have a first part (1 to at most 3 points) which sets a student up for a paragraph length reply to a second part, which would be the paragraph .)  Student  response  expected     Data table shows speed is increasing less and less, I expect speed will increase by 0.5 km/h since increase in each time is halved. Could discuss in terms of decreasing average acceleration instead. Could mention that speed is approaching a limit or a maximum. Could make a sketch graph showing increase in speed leveling off. MATH concepts – taking differences, series pattern, approaching a limit. Representation – observe math pattern – translate into verbal form

Possible  paragraph  length  response  question  –  part  2 b) In a coherent paragraph-length response, which may include diagrams and equations

as appropriate, explain the physics that accounts for the motion of the barge.

Example of appropriate student scratch work. Note that this has all the aspects of a usual solution that might have been expected on an AP Physics B problem. If this has been done, then ordering the pieces by numbering them as in the list below would be the next step in student thinking

Ordered List – would be done by simply putting a number on each part of the scratch work. 1. Graph: increasing v 2. Graph: decreasing a (slope of v) 3. Free body diagram – tension and drag 4. Newton’s 2nd law: FT – FD => a 5. FT and m constant – reason for conclusion steps 6. Larger drag => smaller a 7. Therefore drag increases with speed

Paragraph  example  part  b  –  two  acceptable  solutions    “Long”  version  –  5  or  so  sentences  

From the information in the table, the barge is speeding up but the speed increases less during each interval of time. This means that as the barge is accelerating, the acceleration is decreasing with time. Since acceleration is caused by the combination of all the forces exerted on the barge, the net force must be decreasing with time. The tugboat exerts a constant tension force on the cable, which exerts a constant tension force forward on the barge, there must be a force in the opposite direction, which increases as the barge speed increases. There must be a drag force exerted by the water on the barge that increases as the speed of the barge increases.

“Terse”  version    –  3  sentences    

The table shows that barge speeds up at a decreasing rate as time goes on, so barge must have decreasing acceleration. By Newton’s second law this is caused by a decreasing net force on barge. Since forward tension force is constant, I conclude that there must be a backward drag force that increases as the speed through water increases.

(Solutions could include sketch graphs, free body diagrams and equations from the analysis.)