Abstract

Objective

The objective of this project is to determine the relation between elapsed time and distance traveled when a moving object is under constant acceleration.

Introduction

You know from experience that when you ride your bike down a hill, it's easy to go fast. Gravity is giving you an extra push, so you don't have to do all the work with the pedals. You also know from experience that the longer the hill, the faster you go. The longer you feel that push from gravity, the faster it makes you go. Finally, you also know that the steeper the hill, the faster you go.

The maximum steepness is a sheer vertical drop—free fall—when gravity gives the biggest push of all. You wouldn't want to try that on your bicycle!

In free fall, with every passing second, gravity accelerates the object (increases its velocity) by 9.8 meters (32 feet) per second. So after one second, the object would be falling at 9.8 m/s (32 ft/s). After two seconds, the object would be falling at 19.5 m/s (64 ft/s). After three seconds, the object would be falling at 29 m/s (96 ft/s), and so on.

Measuring the speed of objects in free fall is not easy, because they fall so quickly. There is another way to make measurements of objects in motion under constant acceleration: use an inclined plane. An inclined plane is simply a ramp. You're making a hill with a constant, known slope. With a more shallow slope, the acceleration due to gravity is small, and the object will move at a speed that is more easily measured.

This project will help you make some scientific measurements of the "push" from gravity, using a marble rolling down an inclined plane. You'll record the experiment with a video camera, and use frame-by-frame playback to analyze the motion over time.

Terms, Concepts, and Questions to Start Background Research

To do this project, you should do research that enables you to understand the following terms and concepts:

• velocity,
• acceleration,
• inclined plane,
• mass,
• gravity.

Questions

• What is the formula for velocity as a function of time when an object is subject to constant acceleration?
• What is the formula for distance traveled as a function of time when an object is subject to constant accleration?

Bibliography

• For background information on inclined planes, see these references:
  • Wikipedia contributors, 2006. "Inclined Plane," Wikipedia, The Free Encyclopedia [accessed September 25, 2006] http://en.wikipedia.org/w/index.php?title=Inclined_plane&oldid=79997799.
  • Henderson, T., 2004. "Inclined Planes," The Physics Classroom, Glenbrook South High School, Glenview, IL [accessed September 25, 2006] http://www.glenbrook.k12.il.us/gbssci/Phys/Class/vectors/u3l3e.html.
  • Duffy, A. "Inclined Plane," Boston University, Interactive Physics Demonstrations [accessed September 25, 2006] http://physics.bu.edu/~duffy/semester1/c5_incline.html.
• If you want to make a fancier set-up to release a steel marble electrically, see the following reference:
  U.C. Regents, 1996. "Acceleration," U.C. Berkeley Physics Lecture Demonstrations [accessed September 25, 2006] http://www.mip.berkeley.edu/physics/A+0+20.html.

Materials and Equipment

To do this experiment you will need the following materials and equipment:

• inclined plane:
  • You'll need a flat board, about 2 m long (make sure that you can fit the whole track in the field of view of your video camera).
  • Cut a groove straight down the middle to guide the rolling marble, or glue a straight piece of wood along the length of the board to act as a guide.
  • Mark a starting line across one end.
  • Mark a distance scale on the edge of your inclined plane. Use alternating light and dark colors every 5 cm. For better accuracy at early time points, alternate every cm for the first 10 cm. Make sure that the distance scale is clearly visible in your video recording.
  • You will also need some wood blocks (about 2.5 cm thickness) to raise up one end of the board.
• tape measure (for measuring height and length of inclined plane),
• marble,
• helper,
• video camera:
  • You must be able to do frame-by-frame playback of the video recording.
• tripod,
• plastic wrap (e.g., Saran wrap) and marker for tracking position on screen when playing back recordings.

Experimental Procedure

In this experiment, the goal is to measure the distance the marble travels in equal time intervals as it rolls down an inclined plane.

1. Set up your inclined plane on a single block, so that it has a low slope. If the slope is too high, the marble will roll too fast, and it will be too hard to make accurate measurements.
2. Set up the video camera on the tripod so that the inclined plane nearly fills the viewfinder. Make sure that you can clearly see the distance scale on the side of the inclined plane. You will need this for making your distance measurements when you play back the video recording. If your camera has an on-screen timer, turn it on for help in timing your experiment during playback.
3. Hold a marble in place at the starting line.
4. Have your helper start recording with the video camera.
5. Say "1, 2, 3, go" and release the marble as you say "go", being careful not to give it a push. Your "go" will mark the starting point when you play back the recording.
6. Repeat this at least 5 times.
7. Play back the recording, and find a frame before you have let go of the marble. Mark the starting position (use plastic wrap to protect the monitor screen).
8. Advance the recording frame by frame until the frame where you let go of the marble. This will be time t = 0.
9. Advance the recording by a single frame. (Most video cameras record 30 frames per second, so the elapsed time between each frame is 1/30 of a second.)
10. If the marble has moved a discernible amount from the starting position, measure and record the new position. If not, advance by another frame. Keep track of each frame advance, because this is how you will keep track of time.
11. Continue to measure the position as the marble rolls down the inclined plane. For each measurement, record the distance traveled (from the starting line) the elapsed time.
12. Repeat the measurements, at the same time points, in each of the five recordings you made.
13. Calculate the average and standard deviation for the distance the marble has traveled at each time point.
14. Graph the average distance traveled (y-axis) vs. time, in terms of the number of video frames (x-axis).
15. Graph the average distance traveled (y-axis) vs. time squared. Compare the two graphs.

Variations

• Does the mass of the marble affect its acceleration? Try the experiment with marbles of different masses. Or, even better, compare a steel marble (e.g., a pinball or a large ball bearing) with a glass marble of the same diameter. Use a gram balance to weigh each marble (you can always weigh them at the post office if you don't have a gram balance available at home or school). Is the acceleration the same or different for the marbles with different masses?
• For another method of measuring distance traveled and velocity of an object rolling down an inclined plane, see the Science Buddies project, Distance and Constant Acceleration.
• Use your measurements to calculate the approximate velocity of the marble at each selected time points as it rolls down the inclined plane. Measure the distance the marble has traveled from the starting line in two adjacent frames (f_n and f_n+1). Subtract the distance measured in frame f_n from the distance measured in frame f_n+1. Divide by the result by the time elapsed between two frames (1/30 of a second). Repeat this process for several different pairs of frames from the beginning, middle, and end of the recording. Make a graph of velocity (y-axis) vs. time (x-axis). Does this graph look more like the distance vs. time or the distance vs. time squared graph?
• What happens if you change the slope of the inclined plane?
• One reason a marble was chosen for this experiment was to minimize the frictional forces which counteract the acceleration of gravity. Try repeating the experiment with other rolling objects (e.g., a toy car with the same mass as the marble) or different surface treatments (e.g., smooth, waxed surface, vs. rough, sandpapered surface). Can you detect a decrease in acceleration due to increased friction?
• For more advanced students:
  • If you have studied trigonometry, you should be able to derive a formula that describes the acceleration, a, of the marble as function of the angle, θ, of the inclined plane (see Henderson, 2004).
  • If you have studied calculus, you should be able to explain both velocity and acceleration as the first and second derivatives, respectively, of distance traveled with respect to time. Conversely, you should be able to explain velocity and distance traveled at a given time as the first and second integrals, respectively, of acceleration with respect to time.

• For more science project ideas in this area of science, see Physics Project Ideas.

Credits

Andrew Olson, Ph.D., Science Buddies

Sources

This project is based on:

• Masukawa, J.M., 2003. "Relation between Acceleration and Angle of Inclination," California State Science Fair Abstract [accessed September 25, 2006] http://www.usc.edu/CSSF/History/2003/Projects/J1524.pdf.
• http://physics.clarku.edu/courses/120labs/Lab2.pdf

Last edit date: 2006-10-13 21:00:00