Speedometer

To measure a velocity, try this easy recipe:
One part distance, one part time,
One part direction, one part rhyme.

Other Items You May Need:

butcher paper, or very large box and scissors graph paper
ruler
yardstick
tape
calculator with a square root function
marker
a friend

Challenge: Spotting exactly where the ball lands.

Build the course shown below:

Hint: Try this activity with a friend. Take turns releasing the ball. Before your friend releases the ball, fix your eyes on the landing area. Don't watch the ball descend the course. Draw your X as soon as you see the ball land on the paper.

Here's what to do:

1. Assemble the framework shown above.

2. Cut a piece of butcher paper at least 3 1/2 feet long. Tuck one end of the paper under the right end of the framework. Leave about 3 1/2 feet of paper sticking out.

3. Use a yardstick to draw a straight line across the length of the paper starting about 5 inches away from the right edge of the framework. Then draw a tick mark on the line each inch from the end of the paper nearest the framework to the far end of the paper - so the paper has its own built-in ruler.

4. Attach the six adjustable supports to the framework. When you have finished, it should look like the picture above. Don't worry about exact position now. You can adjust the supports as you attach track.

5. Now connect the pieces for the course: 5 long straights, 1 flex-track and 1 short straight. The track connects like this:

6. Work with your friend to position the course in the framework. The flex-track and the short straight piece at the bottom of the course should extend about 5 inches beyond the framework. Attach the course to the six adjustable supports.

7. Check the ends of the course. The top end of the course should slant the same way as the rest of the long straights in the course. The bottom end of the course should be level. Adjust the flex-track if you need to.

8. Now choose five starting points on the course. You're going to want to look carefully at your data later, so choose regular intervals - every 10 inches from the bottom end of the course. Mark these starting points with tape and label them. For each starting point, use a letter and its distance from the end of the course. (Label the point closest to the bottom of the course point A, the next one point B, etc.)

9. Now adjust the butcher paper so that the start of the measuring line is even with the end of the short straight track.

10. You're done building. Now start collecting data, using the chart below. Work together with your friend, taking turns releasing the ball, observing and measuring.

Starting point A B C D E
Ramp Length          
Prediction:
Distance ball shot past end of ramp.
      Prediction: Prediction:
Trial 1: Distance ball shot past end of ramp.          
Trial 2: Distance ball shot past end of ramp.          
Trial 3: Distance ball shot past end of ramp.          
Average distance ball shot past end of ramp.  (Trials 1-3)          
Average distance ball shot past end of ramp squared.          

Try the experiment releasing the ball at point A three times. Record the results in column A of the chart. Then figure out the average of the three tries. First add the distance the ball shot past the end of the ramp for Trials 1, 2 and 3. Then divide the sum by 3. (You may need pencil and paper, or a calculator.) Record the result in column A of the chart.

Repeat this experiment at points B and C. Make sure to record your data in the chart.

For now, leave the last two columns (starting points D and E) blank.

To fill in the row labeled Average Distance Ball Shot Past End of Ramp Squared on the data chart, you will need to perform another simple calculation. For each starting point, A, B, and C, multiply the average distance the ball shot past the end of the ramp by itself. (You may need pencil and paper, or a calculator.)

11. Now graph the data from the Ramp Length row and the Average Distance Ball Shot Past End of Ramp Squared row. Graphing will give you a better look at any interesting patterns that may be hidden in your data. Use the graph paper. Draw a straight line that connects all three points as possible and extend this line up and to the right.

12. Now use your graph to make two predictions about what will happen when you place the ball at starting points D and E. To make your prediction, you will need a calculator to help you convert the information in your graph back to something you can use easily in your table. First find the Average Distance Ball Shot Past End of Ramp Squared for starting points D. Put this amount into the calculator. Then push the square root key. Ask an adult if you don't know which key this is. Write this number in the column for starting point D as your prediction for Distance Ball Shot Past End of Ramp. Then do the same thing for starting point E.

13. Now test your predictions for point D. Release the ball from starting point D. Observe and measure. Record your results in column D of the chart. Did you predict the distance the ball traveled beyond the ramp?

14. Release the ball from starting point E. Observe and measure. Record your results in column E of the chart. Did you predict this distance?

15. Think about your data. How can you use your project to measure the speed of the ball - as a speedometer?

So how does your speedometer measure speed? The speedometer you built is even more amazing than a car speedometer, because it has time built in. In scientific terms, you have built an apparatus that controls for time. All speedometer track layouts end with a level course at the same height. (16 inches, for example) As long as the ball leaves the course at that height and parallel to the floor, it will take it 0.289 seconds for the ball to reach the floor no matter where you start the ball on the course! In the absence of air resistance or other forces, gravity accelerates all objects at 32 feet/second per second.

Because your speedometer controls for time, it simplifies calculations for speed. Instead of measuring a speed on every trial, you just measure a distance. It allows you to rewrite the equation:

speed = distance/time
as
speed = distance/0.289 seconds.

So, if a ball travels 26 inches, its speed as it leaves the course is about 90 inches per second, or 7.5 feet per second, or about 5 miles per hour. If you want, you can now go back to your original data table and calculate the speed of the ball when dropped from each point.

Why does the height from the floor determine the time? Falling objects accelerate at the same rate, so the time they take to fall to the floor is constant. The ball's forward motion affects the path it takes to reach the floor, but surprisingly, it does not affect how long it takes the ball to fall. It's not only true of a rolling ball. A bullet shot from a level gun and a bullet dropped from the height of the gun barrel will land at the same time, though they take very different paths to the ground.

Easy Variations

1. Build a steeper track, and start a new data table. To help you make predictions, use the data table and graphs from the first course. After you have tried it, check the results against your prediction. Did the ratio (the Average Distance the Ball Shot Past the End of the Ramp Squared to Ramp Length) change?

2. Repeat Variation 1, but with a shallower track. Did the ratio change this time?

Stage III

Let's explore why the plot of ramp length vs. distance ball shot past end of ramp squared gives a straight line. It probably seems weird that this should be the case, but hopefully it will become clear soon!

Note: In this analysis we neglect the rotational kinetic energy of the ball. It will would need to be included if you wanted to predict the exact slope of the line in the plot. For our purposes we can neglect it.

First, we need to realize that the time it takes for the ball to drop from the end of the ramp to the floor is the same regardless of the speed of the ball. (This assumes that the track is perfectly horizontal where the ball flies off.) This is important.

Let's call the distance the ball shoots past the end of the ramp d. Then,

D = vt

where v is the horizontal velocity and t is the time it takes to fall. (t is a constant)

Now we need to explore v a little more carefully and see how it depends on ramp length.

Neglecting losses due to friction, all of the difference in potential energy of the ball is converted into kinetic energy.

Mgh = (Mv²)/2

Solving for v,

V = Sqrt(2 g h)

Plugging this into our original formula for d

D = t Sqrt(2 g h)

Squaring both sides of the equation gives

D² = (4 g t²) h

But 4 g t² is just a constant and doesn't ever change in our experiment. Also, h is just some constant times the ramp length. So

D² = h * c

Where c is some constant. C determines the slope of the line in our plot. We have shown that the plot should be a straight line!

Exercises

Stage III

1. What will increasing the steepness of the track do to the plot?

Answer: It will change its slope. It will have a lower slope. (It will look more flat.) We can see this by noticing that for the same ramp length the ball will go faster if the track is steeper. And if the ball goes faster the distance will be greater.

2. What will happen to the plot if you change the height the ball has to fall before reaching the ground?

Answer: Changing the height will change the time it takes to fall. This time helps determine the constant, c, which is the slope of the line. So changing the height will change the slope of the graph. How will the slope of the graph change if we decrease the distance the ball falls? (It will get steeper.)

You can use what you learned from Speedometer to help tinker with your own projects, especially those that send a ball flying. It can help you predict just how the ball will fall when it leaves the track.