Topic  1 How can we describe the way things move?

Distance, Speed, and Time

Linear motion describes how objects travel in a straight line.  There are three quantities that we can measure:

• Speed, physics code s, in metres per second - the rate at which it covers distance;

• Distance, physics code d, in metres;

• Time, physics code t.

You will have used the familiar equation:

In triangle form:

In these notes we will use SI units (m/s) although you can use the formula with km and hours, or miles per hour.  Some useful conversions are:

• 1 km/h = 0.28 m/s

• 1 mph = 0.44 m/s

Question 1.  A runner completes a 100 m race in 9.8 seconds.  What is her average speed?  ANSWER

Velocity is speed in a given direction, e.g. 4 m/s from left to right.  This example will show the difference.  Suppose the runner travels 63 m by running round the track.  If you look at the straight line distance between A and B, you will see that it is 40 m.  The runner's velocity will be given by

velocity = straight line distance from A to B ÷ time

Question 2:  A runner runs half way round the race track above.  He runs a total distance of 63 metres at a speed of 8 m/s.

(a) What time does he take?

(b) What is his velocity from A to B?  (It is NOT 8 m/s)    ANSWER

Velocity is NOT a posh word for speed.

Acceleration

Acceleration is the rate at which speed changes:

• When objects go faster, the acceleration is positive;

• When objects slow down, their acceleration is negative (deceleration).

The units for acceleration look strange - metres per second squared (m/s²).  It is easier to say this as "metres per second every second".

The equation that links speed and acceleration is:

In triangle form:

The physics codes are:

• a - acceleration

• t - time

• Ds - change in speed.  The triangular symbol D is "Delta", a Greek capital letter 'D', which means "change in".  You will see this used a few times at GCSE, but more at A level.

The change in speed can be worked out easily:

Change in speed = speed at end - speed at start

Worked Example

Question 3:  A car accelerations from 20 m/s to 40 m/s in 15 s.  What is its acceleration?  ANSWER

If the speed at the end is lower than the speed at the start, the acceleration is negative.

Question 4:  A car slows down from 30 m/s to 15 m/s in 10 s.  What is its acceleration?

ANSWER

Motion Graphs

We can represent the movement of objects using a graph, usually plotting time on the x-axis (horizontal) and the speed or distance on the y-axis (vertical).

Distance-Time Graph

Let us have a look at a distance time graph of a cyclist:

Between A and B the cyclist is cycling at a constant speed.  The gradient (slope) of the graph gives us the speed.  Since the gradient is a straight line, the speed is constant.

Question 5:  What is happening between B and C, and C and D?  ANSWER

Question 6:  How would you tell from a distance-time graph that an object was accelerating?  ANSWER

Speed-Time Graph

The graph below shows a train accelerating from a station along a straight and level track to a maximum speed and slowing down to a stop at the next station.  The easiest way to show this is with a speed time graph.

Acceleration is the gradient of the speed-time graph.

From the graph,

• between O and A, the train is accelerating;

• between A and B, the train travels at a constant speed;

• between B and C, the train slows down.  Slowing down can also be called negative acceleration, or deceleration.  It is given a minus sign.

Distance is the area under the speed-time graph.  To work out the total distance, we would add the areas of:

• triangle OAX;

• rectangle ABXY;

• triangle BCY.

At GCSE acceleration is uniform, which means that the speed [velocity] is changing at a constant rate.    This is shown by a straight line on a speed time graph. 

A tachograph on a lorry is a speed time graph.  It is a very accurate speedometer (in kilometres per hour) which plots a speed time graph on a small circular piece of paper.

The instrument plots a trace which you can see on the card.  There are strict rules on the hours that drivers of buses and lorries may work.  The tachograph records when the driver is driving, is at work not driving, is taking breaks, and resting.  The tachograph is not new; instruments for recording speed have been used on railways for over a century.

For more, visit http://www.digitaltachograph.gov.uk/

Working Out Speed or Acceleration from A Motion Graph (HT ONLY)

You may well have to plot a distance time graph or a speed time graph.  Click HERE for some basic guidance on drawing graphs.

To work out the distance from a speed time graph, you need to work out the area.  If the graph consists of rectangles and triangles, it's simple.  Work out the area of each shape and add them together. 

If the graph is complicated, you can get a good idea of the area by counting the squares.  If more than half a square is under the graph, count it.  If less than half, ignore it.  Generally you get pretty close to the area. 

(There is a mathematical trick called calculus that allows us to work out the area under a graph without counting squares, but that is way beyond what you need to know.)

Question 7 Use the information below in the question.  The train's motion is that described in the graph above.

• The maximum speed of the train is 25 m/s

• The time interval OX is 45 s

• The time interval XY is 45 s

• The time interval YC is 20 s         

What is:

a)      The acceleration between O and A

b)      The acceleration between B and C

c)      The distance covered while the train is at constant speed

d)      The total distance.

     e)   The average speed?  

ANSWER

To work out the gradient of a graph:

• Draw a large rise and a large run:

• Gradient = rise/run

Question 8.  What is the acceleration of the car whose motion is shown in the graph below?  ANSWER

Now try the Topic Quiz

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