Tutorial 2: Interpreting Graphs and Equations

Tutorial 2: Interpreting graphs and equations

I.Calculating by thinking

Interpretations are powerful tools for making calculations. Please answer the following questions by thinking and explaining your reasoning to your partners, rather than by plugging into equations. Space is provided for you to write out your thinking in words.

A.A car travels with constant velocity 20 mph (miles per hour) until 3:50:00 and then speeds up as shown on the graph at right. The driver notices that it takes 3 seconds to speed up from 20 mph to 50mph.

1.When does the car have a velocity of 75 mph?

2.How fast is the car going at 3:50:02?

3.Can you determine how far the car traveled between 3:49:55 and 3:50:00?

4.Can you determine how far the car traveled between 3:50:00 and 3:50:05?

5.Calculate the slope of the inclined section of the graph at right.

i.What is the name of this number?

ii.What is the interpretation of this number? (To jog your memory, the interpretation of a number tells you what that number means physically. Interpretations often start with “It is the number of…”)

B.A boat is speeding up with constant acceleration. The boat speeds up from 10 mph to 22 mph in 3 seconds. A student who is studying this motion subtracts 22 – 10, obtaining 12. How would you interpret the number 12?

C.An object moving with constant acceleration has a velocity of 6 mph at t = 0 and a velocity of 30 mph at t = 6 s. Find the velocity at t = 4 s. Try to do this by talking it through, rather than by writing anything algebraic. Write out your thinking below for future reference.

Consult a TA before you proceed.

II.Interpreting motion graphs

A.At right is the velocity versus time graph for a particular object.

1.What does the single point A indicate about the motion?

2.Give an interpretation of the length labeled c.

3.Give an interpretation of the length labeled d.

4.Give an interpretation of the ratio c/d.

B.Suppose a small ball rolling along a track produced the motion represented on the graph at right. What might the track have looked like? Sketch an arrangement of tracks you might set up in lab to produce that motion.

C.Design an arrangement of tracks to produce the motion represented on the graph at right. Sketch the arrangement.

D.Act out the three different motions represented on the following graphs by sliding your finger along the table. Practice until you can do this without any pauses that are not shown on the graphs.

When you are able to execute the motions and explain how you knew to move as you did, check with a TA.

E.Physics courses, including this one, make a big deal of motion graphs. But why? Three students discuss the reason for studying motion graphs in physics class.

Marcus:We need to be able to interpret motion graphs because it’s part of physics, just like solving equations or making observations. That’s why graphs are part of the assignment.

Bob:I think the graphs are primarily for us, so that we can see visually when the motion is steady or increasing or whatever. They help us understand what’s happening.

Jess:I think the graphs are there to confuse us! That way, the only way we can get everything right is if we really understand it. Like, I keep getting position and velocity graphs confused, but if I actually am careful and think about it, I can get them right.

In a few sentences, describe to what extent you agree and/or disagree with each student.

III.Checking algebraic expressions by thinking

The point of this section is to show that sound interpretations are the basis of sound algebraic expressions – and lousy, or absent, interpretations are the cause of many errors.

A.Here is a sample physics problem that an enterprising student found online:

A uniformly accelerated object has an initial position of 10 cm. The object is at position 24 cm after 2 seconds. How much longer will it take the object to travel 70 cm farther?

The student’s solution to this problem is below.

∆x1/∆t1 = ∆x2/∆t2

14/2 = 70/∆t2

∆t2 = 10 seconds

1.What is the interpretation of the number 7 in this context, if there is one? (7 = 14/2) (Feel free to give the name of the number, also, but be sure to give the interpretation as well.)

2.What is the interpretation of the expression ∆x2/∆t2, if there is one?

3.Is the student’s solution correct? If not, what error would you say the student made? (Let’s assume it’s a mistake in the student’s thinking rather than a careless mistake. What should the student be alert to so as not to make the same mistake again?)

B.Here is another sample physics problem and student solution.

An object moving with uniform acceleration increases speed by 60 cm/s in 2.0 s.
How much longer will it take to increase speed by another 100 cm/s?

∆v1/∆t1 = ∆v2/∆t2

60/2 = 100/∆t2

∆t2 = 3.3 seconds

1.What is the interpretation of the expression on the left side of the first line of the solution? What is the interpretation of the expression on the right side?

2.Was the student justified in setting those two expressions equal to one another? Why or why not?

3.Is the student’s solution correct? If not, what error would you say the student made? (Again, try to identify where the student’s thinking went astray, if anywhere.)

C.Here is one more.

Two runners ran a race between two trees. The first runner had a constant speed of 8.0m/s and took 50.2 s. If the second runner had a constant speed of 7.8 m/s, what was her time?

v1/t1 = v2/t2

8.0/50.2 = 7.8/∆t2

∆t2 = 48.9 seconds

1.Interpret the left and right sides of the first equation. Are they really equal? Are they relevant?

2.Is the solution correct? If not, what error would you say the student made? (Again, try to identify where the student’s thinking went astray, if anywhere.)

D.One of the big ideas in this tutorial is making sense of algebraic expressions. Consider the following conversation among three physics students who are trying to do their homework.

Alan:The most efficient way to solve physics problems is to find the right equation and plug in the right quantities.

Beatriz:It’s no use being fast if you end up being wrong. The best way to solve a physics problem is to think it through in a way that makes sense. Then, if you still have to do any equations after that, you can check that they agree with your thinking.

Carly:It’s no use thinking if it doesn’t get you anywhere. You shouldn’t expect physics problems to make sense all the time; sometimes, you need to just use the equations.

Which student do you agree with the most? Explain your thinking.

Consult a TA if you reach this point.