Finding average acceleration with a ticker-timer
This a practical approach to acceleration and its definition.
Apparatus and materials
Elastic cords for accelerating trolley
Rod for attaching elastic cord to trolley
Ticker-timer with power supply unit
Health & Safety and Technical notes
If trolley runways are used, remember that one is too heavy for a single adult to manipulate and carry safely.
a Thread a length of ticker-tape through a ticker-timer and attach the end to a trolley.
b Pull a trolley with a fixed force along a bench. Loop one end of the elastic cord around a rod attached to the trolley. Keep the force constant by making sure that the cord is always stretched by the same amount as the trolley moves. Practice doing this.
c Choose and cut through a dot near to the start of the tape. Do this when the trolley is travelling quite slowly but the dots are far enough apart to clearly distinguish one from another.
d Count ten dot-to-dot spaces and cut the tape, through a dot, again. You have cut a 'ten-tick-tape'.
e Count 40 more dot-to-dot spaces along the tape. Then cut the next 10 dot-to-dot spaces to make another ten-tick-tape.
f Draw a horizontal line, as a time axis, on a piece of paper. Glue your tapes, vertically and 10 centimetres apart, so the bottom of each tape touches this axis. This 10 centimetres represents 1 second.
g Draw a vertical axis anywhere to the left of the first tape. This is a velocity axis.
h Mark a scale, in centimetres per second, on your vertical axis. Each vertical centimetre on your axis represents 5 centimetres per second.
i Use your velocity axis to help you to work out the first velocity and the second velocity. You can call these u and v. (Remember that u comes before v in the alphabet, just as the first velocity comes before the second.)
Work out the average acceleration of your trolley during the time between the two tapes. Acceleration is 'rate of change of velocity'. It is equal to the change in velocity divided by the time.Average acceleration = change in velocity/time taken.
The change in velocity is the difference between the two velocities, v - u. The time between these two tapes is t, which in this case is 1 second.
j Draw a straight line between the centres of the tops of the tapes. Draw a horizontal line from the centre of the top of the first tape. Draw a vertical line through the top of the second tape. You have made a right-angled triangle. The length of the base of your triangle, using the units of the horizontal axis, is 1 second.
k Find the gradient of the line connecting the tops of the two tapes. Measure the height of the triangle, using the units of the velocity axis, which are centimetres per second (cm/s). Divide the height of the triangle, in centimetres per second, by the base, in seconds. This gives you the average acceleration in centimetres per second per second (cm/s2).
1 This introduces students to mathematical modelling of accelerated motion. Many students find the concepts challenging, and the introduction in a practical context is very helpful.
2 If your ticker-timers use a carbon disc, tell students to be sure to pass the tape under the disc. If you are using photosensitive tape, explain that they must wait a few minutes after a run for the dots to appear.
3 The whole procedure above assumes that the ticker-timers produce 50 dots per second (mains frequency of the AC voltage driving them). If your ticker-timer produces more dots per second (some produce 100 dots per second) you will have to change the text.
You will need to explain that the ticker-timer makes dots at regular intervals. You could say:
Each period between dots is a 'tick' of time. A low voltage alternating at the same frequency as mains (50 Hz) drives the ticker-timer. If the ticker-timer produces 50 dots per second, then a tick is equal to 1/50 or 0.02 seconds.
4 Step h of the procedure: The length of each tape is the number of centimetres that the trolley travelled in a fifth of a second, or 0.2 seconds. So students could multiply the length by 5 to find the velocity of the trolley in centimetres per second. The actual lengths of the tapes are fixed proportions of the velocity.
5 Step i of the procedure: The dot at the start of second tape that students make is 50 spaces away from the very first dot they chose. The time between the first dot of their first tape and the first dot of their second tape is 1 second. The time between the middle of their two tapes is also 1 second. You may need to explain this.
An alternative, which may help students to understand the concept of acceleration, is to use mixed units for time. Ask them to calculate u and v in cm/s, but describe the time between these tapes in 'ten-ticks'. This will give units of acceleration as 'cm/s per ten-tick'.
6 The procedure says nothing about the distinction between speed as a scalar quantity and velocity as a vector quantity. But since acceleration is a vector quantity we have chosen to write in terms of velocity.
7 The nature of the motion in the period between the two tapes makes a good discussion point. Students should imagine the intervening tapes. Will the tapes increase steadily in length, or not? If they do not, then the gradient of the velocity-time graph varies, and acceleration varies. The intervening tape could be cut into 10 tick lengths and stuck at the appropriate position on the graph to see if their prediction was correct.
If the tape increases steadily in length, then the gradient is constant, and the actual acceleration always matches the sloping line which students have drawn on their graphs. Acceleration is then constant, or uniform. And in that case average acceleration, and acceleration at all times during the motion, are the same. Then:
acceleration, a = (v-u)/t.
This experiment was safety-checked in January 2005
Daniel McCormack : Mon 2 July 2007: Sticking tentick lengths of tickertape onto a sheet of paper seems a rather silly way of drawing a velocity-time graph. Why not measure the length of each piece of tape, calculate the velocity, and plot a proper graph?
Julie Sutcliffe : Tue 14 October 2008: This was great for all abilities. Using the ticker tape is not silly; it allowed the pupils to see that the distance changed.