**Speed, Distance and Time**

The following is a basic but important formula which applies when speed is
constant (in other words the speed doesn't change):

Speed = __distance__

time

Remember, when using any formula, the units must all be consistent. For example
speed could be measured in m/s, distance in metres and time in seconds.

If speed does change, the average (mean) speed can be calculated:

Average speed = __total distance travelled__

total time taken

*Example*:

If a car travels at a speed of 10m/s for 3 minutes, how far will it travel?

Firstly, change the 3 minutes into 180 seconds, so that the units are
consistent. Now rearrange the first equation to get distance = speed × time.

Therefore distance travelled = 10 × 180 = 1800m = __1.8km__

**Units**

In calculations, units must be consistent, so if the units in the question are
not all the same (e.g. m/s, m and s or km/h, km and h), change the units before
starting, as above.

The following is an example of how to change the units:

*Example*:

Change 15km/h into m/s.

15km/h = 15/60 km/min
(1)

= 15/3600 km/s = 1/240 km/s (2)

= 1000/240 m/s = __4.167 m/s__ (3)

In line (1), we divide by 60 because there are 60 minutes in an hour. Often
people have problems working out whether they need to divide or multiply by a
certain number to change the units. If you think about it, in 1 minute, the
object is going to travel less distance than in an hour. So we divide by 60, not
multiply to get a smaller number.

**Velocity and Acceleration**

Velocity is the speed of a particle __and__ its direction of motion
(therefore velocity is a vector
quantity, whereas speed is a scalar quantity).

When the velocity (speed) of a moving object is increasing we say that the
object is *accelerating*. If the velocity decreases it is said to be
decelerating. Acceleration is therefore the rate of change of velocity (change
in velocity / time) and is measured in m/s².

*Example*:

A car starts from rest and within 10 seconds is travelling at 10m/s. What is its
acceleration?

Acceleration = __change in velocity__ = __10__ = __1m/s²__

time
10

**Distance-time graphs**:

These have the distance from a certain point on the vertical axis and the time
on the horizontal axis. The velocity can be calculated by finding the gradient
of the graph. If the graph is curved, this can be done by drawing a chord and
finding its gradient (this will give average velocity) or by finding the
gradient of a tangent to the graph (this will give the velocity at the instant
where the tangent is drawn).

**Velocity-time graphs/ speed-time graphs**:

A velocity-time graph has the velocity or speed of an object on the vertical
axis and time on the horizontal axis. The distance travelled can be calculated
by finding the *area* under a velocity-time graph. If the graph is curved,
there are a number of ways of estimating the area (see trapezium rule below).
Acceleration is the gradient of a velocity-time graph and on curves can be
calculated using chords or tangents, as above.

On travel graphs, time always goes on the horizontal axis (because it is the independent variable).

**Trapezium Rule**

This is a useful method of estimating the area under a graph. You often need to
find the area under a velocity-time graph since this is the distance travelled.

Area under a curved graph *=* ½ × d × (first + last + 2(sum of rest))

d is the distance between the values from where you will
take your readings. In the above example, d = 1. Every 1 unit on the horizontal
axis, we draw a line to the graph and across to the y axis.

'first' refers to the first value on the vertical axis, which is about 4 here.

'last' refers to the last value, which is about 5 (green line).]

'sum of rest' refers to the sum of the values on the vertical axis where
the yellow lines meet it.

Therefore area is roughly: ½ × 1 × (4 + 5 + 2(8 + 8.8 + 10.1 + 10.8 + 11.9 +
12 + 12.7 + 12.9 + 13 + 13.2 + 13.4))

= ½ × (9 + 2(126.8))

= ½ × 262.6

= __131.3 units²
__

© Matthew Pinkney