Topic 9.3
Terminal speed
Investigating terminal speed
Aims
The terminal speed of an object falling in a fluid is the speed that the object reaches when the
drag force on it (due to the fluid) is equal and opposite to the object’s weight. In this
investigation, steel balls of different diameters are used to investigate how the terminal speed
of a ball falling in a fluid depends on the diameter of the ball.
Safety
 Ensure the cylinder of liquid is in a tray in case the liquid spills.
 Wear suitable protective gloves.
 Take care to ensure the liquid does not splash on your clothing or any part of your body.
Equipment, materials and method
1 For this experiment you will need at least 5 steel balls of different diameters less than
5 mm; a tall transparent cylinder, fitted with three rubber bands, standing in a tray; a top
pan balance; a micrometer; a stopwatch or stopclock; a metre ruler; a stand fitted with a
boss and clamp; a set square; suitably viscous fluid (for example, glycerol); a magnet (to
retrieve each ball from the bottom of the cylinder); paper towels and protective gloves.
2 Place the rubber bands as markers horizontally around the cylinder and position them so
that one is near the fluid surface, one is near the bottom of the cylinder, and one is
midway between the other two. Use this arrangement to check that the speed of the ball is
the same in the bottom half of the cylinder as it is in the top half. If so, the speed can be
assumed to be constant as the ball falls through the fluid. Record your measurements
including the precision of each measurement.
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Topic 9.3
Terminal speed
3 Measure the mass m and diameter d of each ball. Make suitable measurements to
determine the terminal speed as it falls through the fluid. Record all your measurements in
a suitable table.
Theory
The ball in the fluid is acted on by a drag force D due to the viscosity of the fluid, which acts
in the opposite direction to the force of gravity on the ball (that is, its weight W).
Assuming the drag force D is directly proportional to the diameter of the ball d and to its
terminal speed v, it can be shown that
W= k v d
where k is a constant that depends on the density of the ball, the density of the fluid and the
viscosity of the fluid.
Therefore a graph of W against v  d should be a straight line through the origin.
Results
1 For each ball, calculate:
(a) v  d
(b) the weight W
Give the results of your calculations in a second table.
2 Plot a graph of W against v  d
Questions
1 (a) Describe the measurements you made to check that the speed of descent in each half
of the cylinder was the same.
(b) Explain why your measurements show the ball falls at terminal speed between the
upper and lower markers.
(c) (i) What is the most significant source of error in this part of the experiment?
(ii) Discuss how you could reduce the most significant source of error.
2 (a) Discuss whether or not your graph is a straight line through the origin.
(b) Explain how you would use your graph to determine k.
3 Taking account of buoyancy on the ball in the fluid, it can be shown that
k=
3
(1 
0
)

where  is the density of steel,  0 is the density of the fluid and  is its viscosity. Given
that the density of steel is 7800 kg m−3 and given the value of k, describe what further
measurements you would need to make to determine the viscosity of the fluid.
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Topic 9.3
Terminal speed
Technicians Notes
Equipment and materials required
Each group or pair of students will require the following equipment:
 5 steel balls of different diameters less than 5 mm
 a tall transparent cylinder, fitted with three rubber bands, standing in a tray
 a top pan balance
 a micrometer
 a stopwatch or stopclock
 a metre ruler
 a stand, boss and clamp
 a set square
 suitably viscous fluid (for example, glycerol, or shower gel / shampoo)
 a magnet (to retrieve each ball from the bottom of the cylinder)
 disposable gloves
 paper towels
 graph paper
Safety
 Ensure the cylinder of liquid is in a tray in case the liquid spills.
 Provide suitable protective gloves.
AQA Physics A AS Level © Nelson Thornes Ltd 2008
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Topic 9.3
Terminal speed
Teachers Notes
Aims
 To measure accurately the speed of a steel ball falling at its terminal speed in a fluid.
 To relate practical measurements to straight line graph theory.
 To consider further measurements necessary to determine the viscosity of a fluid.
Safety
 Ensure the cylinder of liquid is in a tray in case the liquid spills.
 Remind students to wear protective gloves.
 Ensure the liquid does not splash on clothing or on any part of the body.
Teaching notes
4 The experiment could be carried out by a small group as one of a series of practical
experiments.
5 Ensure the fluid used gives measurable timings with the largest ball.
6 Remind the students to check the zero readings of all their measuring instruments. They
should make several timings using the same ball to test if the ball moves at the same speed
in each half of the cylinder. They will need to consider the mean value of the timing for
each half to decide if the timings are the same.
7 They will need to use the magnet to drag each ball out of the cylinder.
8 Students may need to be taken through straight line graph theory and how the general
equation y = mx + c relates to the given theoretical equation.
Answers to questions
1 (a) Make several measurements of the time taken for the ball to fall from the upper
marker to the middle marker and for an equal distance from the middle marker to the
lower marker. Use the metre ruler to check the distances are the same.
(a) If the mean value ± the uncertainty of the upper timing overlaps with the mean value
± the uncertainty of the lower timing, the timings can be considered to be the same.
Since the distance for each timing is the same, their average speeds must be the same.
The ball therefore must have moved at the same speed throughout each part of the
descent.
(b) (i) The timings because they will be subject to random variations resulting from
human error.
(iii) Make as many timings as possible to give more reliable values for the mean
value and the uncertainty in the time taken for each half of the descent.
2 (a) The graph should be a straight line through the origin.
The gradient of the graph is equal to k. Measure the gradient of the graph by drawing a
large gradient triangle.
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Topic 9.3
Terminal speed
3 The density of the fluid must be determined. This requires measurement of the mass and
the volume of a sample of the fluid, so that the density may be calculated by dividing the
mass by the volume.
Practical assessment opportunities
1 PSA – Selecting and using equipment: 0–3 marks
(Following instructions and group work; organisation and safety assessed at end of
course)
2 ISA preparation:
practical task – practice in recording measurements, calculating mean values and
uncertainties, processing data and plotting a graph
written task Section A and B – opportunities to discuss reliability, to analyse data and
errors and to describe further measurements needed to find the viscosity of the fluid

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