Topic 8.3
Motion along a straight line
at constant acceleration
Equations of motion
Aims
In this activity you will use mathematics to manipulate two equations that describe motion.
You will then be asked to design an experimental method suitable to test one of the
equations. This will enable you to plot a graph of some sample results and test the equation
using your method.
Safety
This is a planning and analysis activity; there are no experimental safety issues.
Equations of motion
The following equations apply to objects that are moving with constant acceleration.
(1) v  u  at
vu
(2) s  
t
 2 
1 State each of the terms in the equations and their units.
2 By substituting equation (1) into equation (2) show that s  ut  12 at 2
3 On the graph below show how the equation s  ut  12 at 2 can be used to show that the
distance is equal to the area under the graph.
4 Rearrange equation (1) to make an expression for t. Substitute this into equation (2) and
simplify to show that v 2  u 2  2as
Planning
The task now is to plan an experiment to test the equation v 2  u 2  2as . You will consider
a free-falling object such as a weighted card passing through a light gate. The independent
variable should be the distance s, and the dependent variable should be the final velocity v.
Consider how you will set up the apparatus and draw a diagram.
1 Explain how the experiment should be carried out. How will you measure v? Describe the
measurements and the calculations that need to be made.
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Topic 8.3
Motion along a straight line
at constant acceleration
2 Describe any precautions that should be taken to ensure the results are as accurate as
possible.
3 Draw a table for your results. Include a suitable range of values for the independent
variable.
Data analysis
1 Consider the equation in the form y = mx + c, where v2 and s are y and x respectively. If
you plotted a graph of this equation, what would the values of the gradient and the
intercept of your graph represent?
2 Use the data in the table to plot a graph of v2 (on the y-axis) and s (on the x-axis). Draw a
line of best fit for your results. Find the value of the acceleration due to gravity g from
your graph.
Distance dropped / m
0.60
0.80
1.00
1.20
1.40
Velocity / m s1
3.37
3.88
4.31
4.75
5.11
Velocity repeat 1
3.42
3.95
4.37
4.71
5.12
Velocity repeat 2
3.44
3.87
4.40
4.72
5.19
Velocity repeat 3
3.40
3.92
4.32
4.77
5.03
Conclusion
By considering the uncertainties in the experiment, comment on whether your results support
the equation v 2  u 2  2as
AQA Physics A AS Level How science works © Nelson Thornes Ltd 2008
2
Topic 8.3
Motion along a straight line
at constant acceleration
Equations of motion
Aims
 To plan an investigation to test an equation of motion, selecting appropriate apparatus and
methods, including ICT.
 To tabulate and process measurement data, using equations and appropriate calculations.
 To plot and use an appropriate graph to verify the relationship between velocity and
distance for a falling object.
 To relate the gradient and the intercept of a straight line to a linear equation.
 To make reasonable estimates of errors using data and graphs and to draw conclusions.
Link to specification
The following “How science works” concepts are practiced in this activity:
C Use appropriate methodology, including ICT, to answer scientific questions and solve
scientific problems
E Analyse and interpret data to provide evidence, recognising correlations and causal
relationships
F Evaluate methodology, evidence and data, resolving conflicting evidence
Safety
This is a planning and data analysis activity; there are no experimental safety issues.
AQA Physics A AS Level How science works © Nelson Thornes Ltd 2008
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Topic 8.3
Motion along a straight line
at constant acceleration
Teacher notes
Equations of motion
3 u initial velocity (m s−1), v final velocity (m s−1), t time (s), a acceleration (m s−2),
s distance (m).
1
vu
t give s = (u + at + u)t, which when
2
 2 
4 Substituting v  u  at into s  
simplified is s  ut 
1 2
at
2
5
Thus, as shown in the graphic, total area under graph = ut +
6 Rearranging v  u  at gives t =
1 2
at
2
(v  u )
a
(v  u ) (v  u )
vu

t gives s 
2
a
 2 
2
2
2
2
Simplifying gives 2as = (v − u ) and hence v  u  2as
Substituting for t in s  
AQA Physics A AS Level How science works © Nelson Thornes Ltd 2008
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Topic 8.3
Motion along a straight line
at constant acceleration
Planning
The distance may be measured approximately from the middle of the card since the light gate
will measure the average velocity as the card passes through. The card should be weighted
and the light gate held in position with a clamp stand.
However, a point for the students to consider is this: when is the card travelling at its
average velocity as it passes through the light gate? It will not be when the middle of the
card passes through the gate, but when it has been passing through the gate for half of its
transition time. Because the card is accelerating, this will not correspond to the middle of
the card. The error here will be bigger for small distances.
The data logger will record the time that the card intercepts the light gate. The velocity is
calculated using this intercept time and the length of the card.
7 No breezes, fans etc. The experiment should be repeated several times for each height.
The card should be weighted so that it falls vertically.
8
Distance / m
0.60
0.80
1.00
1.20
1.40
Velocity / m s−1
Velocity repeat 1
Velocity repeat 2
Velocity repeat 3
Conclusion
The uncertainty can be estimated from the spread of readings. For the sample readings the spread is
about 2%. The value of g is calculated by measuring the gradient of the graph. The gradient = 2g. For
the sample readings g is about 9.3 m s−2, which is about 5% away from the actual value of 9.81.
Students should deduce that this difference is probably due to the air resistance of the card which is a
systematic error and is not identifiable by considering the spread of readings.
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