An exploration using the Downhill Race exhibit to discover basic ideas of force and motion.
Downhill Race exhibit or snack (or slow-moving object down a ramp)
tape measure or ruler
To Do and Notice
Take one wheel and let it roll down the track. Observe the speed of the wheel as it rolls down.
Measure the distance of the entire track. Time how long it takes the wheel to travel that distance. Make a prediction of how long it will take the wheel to travel half of the distance.
Measure and mark the half-way point of the track. Use this as a new starting point for the wheel, and time how long it takes to reach the bottom. Does half the distance take half the time?
Calculate the ratio of the first time to the second. Is that a familiar number?
Measure and time how long it takes the wheel to travel one-fourth of the track distance. What is the ratio between one-half and one-fourth?
(optional) How long will it take to travel one-third of the distance? What is the shape of the curve if you graph distance vs. time?
What’s Going On
You may have noticed that the wheel takes longer than half the time to go half the distance of the track. In fact, the ratio between the whole track to half the track should be close to 1.4 or the square root of 2. This is the same as the ratio between one-half and one-fourth of the track. This is because the velocity of the wheel is not constant during its trip. You know this must be true because the velocity starts at 0 and ends at non-0.
To figure out how the velocity is changing, consider the forces acting on the wheel. The main force affecting the wheel’s motion is gravity, which causes a constant acceleration. When the acceleration is constant, the velocity is directly proportional to time, or
Since the ramp is at an angle and there are frictional and rotational forces that we’re ignoring, the acceleration is some fraction of gravity, but can be represented by a constant.
The distance traveled by the wheel at any moment can be expressed as
but v is constantly changing. Substituting our expression for v, we get
which is the relationship between the things you measured. If you plot your data, do you get something that looks like a parabola? You discovered that the distance an object moves under constant acceleration is proportional to the square of the time!
Downhill Pace - draft
Julie Yu, Exploratorium, 2011