Galileo's Inclined Plane Experiment - Maple Programming Help

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Galileo's Inclined Plane Experiment

Main Concept

Galileo Galilei is considered to be one of the fathers of modern science due to his extensive research in astronomy and physics. One of his greatest contributions  involved accurately measuring the effect of gravity on free falling bodies. Galileo hypothesized that a falling object would gain equal amounts of velocity in equal amounts of time, which meant that its speed increased at a constant rate as it fell. There was one problem, however, in testing this hypothesis: Galileo could not observe the object's free falling motion and at the time, technology was unable to record such high speeds. As a result, Galileo tried to decelerate its motion by replacing the falling object with a ball rolling down an inclined plane. Since free falling is essentially equivalent to a completely vertical ramp, he assumed that a ball rolling down a ramp would speed up in the exact same way as a falling ball would.

 The Conclusion Using a water clock, Galileo measured the time it took for the ball to roll a known distance down the inclined plane. After many trials, he observed that the amount of time it took for the ball to roll down the entire length of the ramp was equal to double the amount of time it took for the same ball to only roll a quarter of the distance. In other words, if you doubled the amount of time that the ball was rolling, it would travel four times as far. Through this experiment, Galileo concluded that if an object is released from rest and gains speed at a steady rate (as it would in free-fall or when rolling down an inclined plane), then the total distance, s, traveled by the object is proportional to the time squared needed for that travel:     The proportionality constant is exactly half of the acceleration a. For a ball rolling down an inclined plane, this acceleration relates to the gravitational acceleration g via   where $h$ and $l$ are the height and length of the inclined plane. The resulting relationship   allowed Galileo to determine the value of the gravitational acceleration $g.$

Using the slider, choose a height for the inclined plane. Click "Roll" to let the ball roll and press "Pause" to stop it at any point. Using the values you observe for time elapsed and distance traveled after each roll, use the lower plot to graph several points on the Distance vs. Time graph. What kind of relationship appears within the data you collected? Does it match the actual relationship shown when you check the box "See Actual Relationship"? Check "Estimate g" to estimate a value for the gravitational constant, g, using your data points. Does this value match what you expected?

Choose "Reset Ball" to have the ball placed at the top of the ramp again, and choose "Clear History" to return the ball to its starting point, as well as erase any points you recorded in the Distance vs. Time graph below.

  Time: Distance Rolled: Distance Rolled vs. Time

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