What it shows
The slope of a distance-time graph at a point is the instantaneous speed, which is the derivative of distance with respect to time. A car on a ramp that speeds up gives a curve whose tangents get steeper.
The science
Average speed over a time interval is distance divided by time. Taking a smaller and smaller interval gives the instantaneous speed, which is the slope of the tangent to the distance-time curve. For uniform acceleration from rest the distance is s = (1/2) a t squared, so the speed is ds/dt = a t, a straight line through the origin. The slope of the tangent to the curve at time t therefore equals a t, and the tangent speed values plotted against time should give a line. This links the idea of the derivative to a real motion.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- A toy car
- A wooden plank or board about 1.5 m long, propped at a small angle
- A metre tape
- A phone for slow-motion video with a clock or timer app
- Chalk to mark each 10 cm along the ramp
- Graph paper
- A transparent ruler
Safety
For this project: Do the experiment on the floor, away from stairs. Keep fingers clear of the car and the ramp end.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Prop the plank at a gentle slope and mark every 10 cm along it with chalk, starting from the top.
Hold the car at the top, release it without a push and film it beside the tape so that both car and timer show on the video.
Play the video frame by frame and write the time at which the car front passes each chalk mark.
Plot distance on the vertical axis against time on the horizontal axis and draw a smooth curve through the points.
At five different times, place the ruler so that it just touches the curve and draw a tangent line. Do not let the tangent cut across the curve.
Measure the slope of each tangent by choosing two far-apart points on it. These slopes are the speeds at those times.
Plot speed against time. It should be close to a straight line through the origin, and its slope is the acceleration.
Check the acceleration from s = (1/2) a t squared using a single point and compare the two values of a.
What you should see
A curved distance-time graph, tangents that become steeper with time, and a speed-time graph that is a straight line, giving an acceleration that agrees with the value from the distance formula.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1What does the slope of a distance-time graph represent?
The speed of the object.
2What is a tangent?
A straight line touching a curve at a point with the same slope as the curve there.
3What is the derivative of s = (1/2) a t squared?
ds/dt = a t.
4Why does the speed-time graph pass through the origin?
The car starts from rest, so the speed is zero at time zero.
Take it further
Increase the ramp angle and show that the acceleration changes with the sine of the angle.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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