Instantaneous and average velocities
You can use a sensor datalogging system to experience the difference between instantaneous and average velocity.
Apparatus and materials
Light beam sensor assembly (source and sensor)
Computer with datalogging software
Runway, with means to produce a uniform slope
Health & Safety and Technical notes
A string tied across the runway will ensure that the trolley does not damage the pulley or a user. Long runways should be handled by two persons.
a Set up the runway so that the trolley can accelerate down it. Set up a source and detector across the runway.
b The detector should be connected to a computer. The computer should be set to record the time during which the beam of light to the detector is interrupted.
c Cut pieces of card of different lengths to attach to the trolley, one at a time. Measure their lengths. The longest should be 25 to 30 centimetres long.
d Tape the longest card to the trolley, so that it can break the light beam. Tape the centre of the card at the centre of the trolley's length.
e Use the computer system to measure the time for the card to move past the light beam. Get the computer to divide the length of the card by this time. The answer is the average velocity of the trolley during that period of time.
f Replace the card by a much shorter one. Again, fix the card centre to the trolley centre. Release the trolley from the same place on the runway. Find out the new time and new velocity. The time is shorter than before. The new velocity is still an average velocity.
g Find out the shortest length of card that will give consistent values of time and velocity.
1 Students should understand the difference between speed and velocity; a scaler and a vector quantity.
2 For the accelerating trolley, its velocity is changing, instant by instant. Students could be invited to think about these problems:
- What information would the computer tell you if you used a card that was as long as the trolley's journey? (Answer: The average velocity for the whole journey, provided that the light beam is set up halfway along the distance of travel.)
- What length of card would you need to measure an instantaneous value of the trolley velocity? (Answer: A card so thin that it is disappearing would, if the system could sense it, provide the value of velocity at the instant that it passes through the beam. Note that an 'instant' is a time that is so small that it is vanishing.)
- What would be the best way to gather data to produce a smooth and accurate graph of velocity against time: should you use short cards or long ones? (Answer: A card of disappearing length used with many sensor systems along the track would produce a set of points, one for each sensor, for a velocity-time graph. Other methods provide only average velocities.)
These ideas relate, of course, to the mathematical concept of infinitesimal values, and hence to the foundations of calculus, without which our understanding of the world would be much narrower than it is.
A good estimate of instantaneous speed is measured by a car speedometer, while the average speed can be calculated knowing the distance travelled and the time taken.
3 As an extension activity, you could use a pair of source-sensor assemblies linked to the same computer to obtain two velocity values and the time between their measurements. Divide the difference in the velocities by this time to work out a value for acceleration.
This experiment was safety-checked in November 2004