Speedometer Experiment

•	Introduction to Speedometer Experiment
•	Predictions
•	The experiment

Introduction to Speedometer Experiment
You are going to observe the speedometer of a car as it coasts, with various initial velocities, between two fixed points.
You will choose a specific path through empty parking lot or along a seldom-used stretch of road. 
You will choose a specific starting point A and a specific ending point B. 
The path must have the following characteristics:
the legal speed limit should be at least 25 miles per hour
the distance from point A to point B is at least .1 mile and not more than .3 mile
the slope of the road should vary significantly between points A and B; don't choose a place where the incline is pretty much unchanging
the maximum speed of the car as it coasts from A to B, starting from rest at A, should be between 10 and 15 mph
there should be one stretch of the path where the slope of the road doesn't change significantly for a distance of about 100 feet; points C and D will be chosen at least 100 feet apart along this stretch
You will coast from point A to point B several times, reaching point A at different velocities.
You will observe and record the car's speedometer while coasting from point A to point B while remaining within speed limits, with the car driven safely and obeying all traffic laws.
There are several important questions to ask about this situation. Predict the answers to these questions before you conduct the experiment.
Predictions
Question 1: How is the final velocity (the velocity at B) related to the initial velocity (the velocity at A)? 
Describe how the steepness of your path changes from point A to point B.
Complete the following table of velocities, as you would predict them:
Velocity at A 0 5 10 15
Velocity at B 10 ? ?? ??
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If you increase your initial velocity, would your final velocity also increase? If you increase your initial velocity by equal amounts, will your final velocity increase by equal amounts?
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If you change your initial velocity at A by 1 mile per hour, will your final velocity at B
increase by 1 mile per hour 
increase by more than 1 mile per hour
increase by less than 1 mile per hour, or
would you get different answers to these questions depending on your initial velocity? 
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If you coast from A to B, starting with velocity 5 mph at A, then coast again from A to B starting at a significantly greater velocity, will the increase in velocities from A to B be 
about the same both times 
greater for the 'run' with the lower initial velocity
greater for the 'run' with the higher initial velocity
sometimes greater for the 'run' with the lower initial velocity, sometimes for the 'run' with the higher, depending on exactly what the higher velocity is. 
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Question 2: Let C and D be the two chosen points, about 100 feet apart, where there is a significant, constant slope.
If you moved 100 feet without speeding up or slowing down, and if you required 5 seconds to move the distance, then at how many feet per second were you moving?
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If you know how far you move at a constant velocity (i.e., without speeding up or slowing down) and how long it takes, then how do you determine your velocity, in feet/second? 
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As your car coasts between C and D, its velocity will very likely change. You will observe and record a distance and a time between these points. You can use this distance and time to calculate a velocity in feet/second, just as you did when the velocity was unchanging. We call this velocity the 'average velocity'. 
Find the average velocity, in feet/second, if a car moves 100 feet between C and D in 4 seconds.
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Will the car's velocity as it moves from C to D ever be greater than the average velocity you just calculated? Will it ever be less? Will it ever be exactly equal? 
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If we average a bunch of numbers, is the average necessarily equal to one of the numbers? If your answer is 'yes', then explain how it is that this is always the case. If your answer is 'no', give an example to prove your assertion.
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Assuming that the car's velocity changed continuously, never staying the same from one instant to the next, was the car's velocity ever actually equal to the average velocity calculated from distance and time? If your answer is 'yes', at what instant or instants were the actual and average velocities the same? If your answer is 'no', explain how the car's velocity could fail to equal, for at least an instant, the average velocity calculated from distance and time.
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Just what do we mean by the velocity at an instant? What do we mean when we say that the velocity changes from one instant to the next? An instant has zero duration. So when does the next instant occur? Can there be an instant between an instant and the next instant?
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Question 3: If the car has velocity 15 feet/second at one point and 25 feet/second at another, then we might conjecture that the average velocity, as obtained using the distance and time information between the points, is 20 feet/second. Comment on the accuracy of this conjecture for each of the following situations. If you conclude that the average velocity in a situation would not be equal to the 20 ft/sec average of the initial and final velocities, explain why you believe so, and say whether the average velocity would be greater or less than 20 ft/sec. In each case consider whether the velocity spends more time nearer 15 ft/sec or 25 ft/sec.
There is a large downhill slope near the first point, followed by a small slope that runs all the way to the second point.
There is a small slope near the first point and a much larger slope near the second point.
The slope between the two points is constant (e.g., as it is between C and D).
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Question 4: Let C and D be the same two points as in #2.
Can the car be moving when the speedometer's needle is still? Can you determine the speed of the car from the speed of the speedometer needle? Exactly what does the speed of the speedometer needle tell you about the car's motion? 
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Will the speedometer needle's motion between points C and D (where there is a constant and significant slope) be
faster when the car is moving faster
slower when the car is moving faster
the same regardless of speed
sometimes faster when the car is moving faster, sometimes slower when the car is moving faster?
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Suppose we know the speedometer readings at point C and at point D, and we also know how much time was required to move between these points.
How could we use this information to arrive at a number that tells us how fast the speedometer needle was moving, in MPH (miles/hour) per second?
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Question 5: Suppose that the speed of the car is observed at 2-second intervals, and that the first four observed speeds are 12, 15, 13 and 17 feet/second. These four speeds define three 2-second time intervals.
What is the best estimate you can make of the distance traveled by the car during each 2-second interval (there are 3 intervals; you should get three answers)? 
What is the total distance traveled, based on these estimates? 
What is the average speed, based on this total distance and on the 6-second time interval required to travel the distance? 
Is the average speed equal to the average of the four given speeds?
Is the average speed equal to the average of the three average speeds you calculated?
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At what average rate, in MPH per second, was the speedometer needle 'moving' during each of the three intervals? 
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Answer these same questions for the following situation, which involves the same speeds as the previous situation:
In the first 2 seconds the car's speed goes from 12 to 15 ft/s. 
During the next 3 seconds the speed drops to 13 ft/s. 
During the next 1 second the speed increases to 17 ft/sec.
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How were your results for the second situation different in nature than those of the first? 
Were the distances moved equal in both cases? 
If you answered 'yes', explain why the distances were identical. If you answered 'no', explain how the distances could the different with the same velocities. 
Was the motion of the speedometer needle identical in both cases? If you answered 'yes', explain why the motion was identical. If you answered 'no', explain how the motion could the different with the same velocities. 
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Question 6: How should the motion of the speedometer needle vary between three slopes, the second twice as great as the first and the third 5 times as great as the first?
On which slope will the speedometer needle move most slowly? 
On which slope will the speedometer needle move most quickly?
The ratio of the second slope to the first will be 2/1. Will the ratio of how fast the speedometer needle moves on the second vs. the first slope be
about 2/1
about 1/2
significantly more than 2/1
significantly less than 2/1? 
What is the ratio of the third slope to the second, and what do you think will be the ratio of how fast the speedometer moves on the two slopes?
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If the speedometer needle moves at 2 mph per second on the first slope, how fast will it move on each of the others? In each case how long will it take the speedometer reading to change by 10 mph? 
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Question 7: Give a plausible set of speedometer readings for each of the following situations. Assume in each case that the length of the path is such that the car attains a speed of 30 mph when coasting from rest, and that the process requires about 20 seconds. Thus, each set of speedometer readings starts at 0 and ends up at 30. You fill in the middle numbers, as they might be at, say, 2-second intervals.
The car coasts from rest down a steep but fairly short hill then along a gentle downward slope for a slightly longer distance. 
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The car coasts from rest some distance down a gentle downward slope then down a steep hill for about the same distance.
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The car coasts from rest down a moderate and constant downward slope.
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The car coasts down each of the above paths, but this time starting out at 30 mph.
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The experiment
Throughout this experiment, you will coast between points A and B, with your transmission in neutral and your foot off of the brake. If either of these conditions is violated at any time between A and B, do not record data for that run.
You will need at least two people to perform this experiment. 
The driver will be responsible only for operating the vehicle as directed. The driver will not do any timing. The driver will look at the speedometer only to attempt to achieve the appropriate initial velocity at point A.
One or more observers will observe and record times and velocities. It is suggested that two observers, one observing times and the other velocities, would be a desirable combination.
Initial and Final Velocities
Using trial and error, find the greatest speed with which you can pass point A without exceeding 25 mph at any time as you coast to point B.
Record the initial velocity (at A) and the final velocity (at B) for this maximum velocity, and record the total time necessary to move from A to B.
Decrease your initial velocity by approximately 2 mph and record velocities at A and B and total time required from A to B.
Repeat again and again, each time reducing initial velocity by approximately 2 mph, until your initial velocity is zero.
Make a table of your data. Clearly label the table, and append a short synopsis of exactly what you did to be sure your data was accurate.
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First let y = final velocity and x = initial velocity. Plot y vs. x on a full-page graph and sketch the 'ideal' curve indicated by this data. Describe the graph in detail.
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This time let y = change in velocity and x = initial velocity. Plot y vs. x on a full-page graph and sketch the 'ideal' curve. Describe the graph in detail. 
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Compare your results to the predictions you made in Question #1. 
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Explain in your own words how the velocity change depends on the initial velocity. Speculate on why it should be so.
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Average velocities
Choose four initial velocities, approximately equally spaced, spanning velocities from rest to the maximum velocity you used above.
For each initial velocity, reach point A at this velocity and coast. Record your velocity at C, your velocity at D and the time in seconds required to move from C to D.
Give your data in table form.
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Determine the distance in feet from C to D, if you have not already done so.
Compare the average of the initial and final velocities with your average velocity between C and D, as computed from distance and time. You will need to make use of the fact that 1 mph = 1 mile / 1 hour = 5280 ft / 3600 sec = 1.47 ft/sec, rounded to three significant figures.
Are your results consistent with a claim that average velocity between C and D for a trial is equal to the average of the velocities at C and at D for that trial? 
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Compare the average of the initial and final velocities for a trial with your average velocity between A and B for that trial, as computed from distance and time, using the four trials in which your initial velocities are closest to the ones you used in this part of the experiment. Make this comparison for each of your trials.
Are your results consistent with a claim that average velocity between A and B for a trial is equal to the average of the velocities at A and at B for that trial? 
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Speedometer motion
Using the information you have obtained, determine how fast the speedometer needle was moving, in mph/sec, as you coasted from C to D.
Is there any clear relationship between how fast the speedometer needle was moving and how fast the car was moving?
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Suppose that the speedometer needle moves at the same rate between C and D regardless of your velocity (this may or may not be so; you are testing the hypothesis here). If you coast slowly between C and D, then coast between these points at twice the speed, how will your changes in velocity between C and D compare? 
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Velocity Chronicles
Coasting from an initial velocity of 10 mph at A, observe the speedometer at regular 3-second intervals until you reach point B. Read the speedometer as accurately as you can, estimating to 1/2 mph (or even better) if possible.
Give your data.
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For each 3-second interval, determine from the above data, as accurately as possible:
the distance moved during the interval
the average rate, in mph/second at which the speedometer needle moved during the interval.
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Based on your results, make a table of y = distance from point A vs. t = clock time, and plot y vs. t on a full-page graph. Describe your graph.
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Based on your results, sketch a graph of how the slope of the path changes with distance. Describe your graph.
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Based on your results, sketch a side-view picture of the path. Describe your sketch.
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