Subject knowledge and
 challenging concepts

In a situation in which motion is changing, there are various quantities that can be measured. For example, when a space shuttle takes off it has a starting velocity (of zero), and over a certain period of time it will travel a certain distance and reach a final velocity. It will have accelerated during this time. The equations of motion allow these variables to be linked. For example, if the shuttle has an acceleration of 20m/s² for the first 5 minutes (having started from rest), the equations will allow us to calculate the distance it has travelled and its velocity at the end of that period of time.

• Some variables are scalars and some are vectors. Scalar quantities such as mass have magnitude but no direction. Vector quantities such as force have both magnitude and direction. This is the distinction between speed and velocity: speed is a scalar quantity but velocity is a vector as it includes reference to direction. This is important in this topic as there may be situations with objects travelling in different directions or with changes in direction.
• Variables such as force and velocity that are vector quantities may be combined in the following ways (it is important to clarify which direction is being regarded as positive).
  • If they are in the same direction, they are added.
  • If they are in opposite directions, they are subtracted.
  • If they are at right angles they are combined by moving one vector (maintaining its length and orientation) so that the tail of one vector touches the tip of the other. The resultant vector goes from the tail of the first vector to the head of the second (see Figure 3).

Figure 3. Combining vectors at right angles

• There are four equations of motion which describe the relationship between five variables relating to an object in motion.
  
  a = acceleration, in metres per second squared (m/s²)
  s = distance, in metres (m)
  t = time, in seconds (s)
  u = initial velocity, in metres per second (m/s)
  v = final velocity, in metres per second (m/s).
  
  Each of the equations links four of these variables:
  
  v = u + at
  s = ut + ½at²
  v² = u² + 2as
  
  s =

  (u + v)t 2
• Therefore, if three of the variables are given, by selecting the appropriate formula, the fourth can be calculated.
• The variables have to be entered using units, such as metres, kilograms and seconds, and students need to be confident and consistent with these. If, for example, the object is in free fall, the acceleration will be that due to gravity, normally taken as 10m/s². This means that distance has to be in metres and velocity in m/s.
• Velocity and acceleration are vector quantities and these need to be reflected with the use of a negative sign if the direction of any of those quantities is reversed. This includes the acceleration of an object that is still travelling in a positive direction but slowing down.� As well as relating to large, spectacular objects such as spacecraft, velocity and acceleration also apply to regular everyday objects such as supermarket trolleys. For example, if a loaded trolley were pushed and allowed to roll until it comes to rest, if its initial velocity and the distance were known, its acceleration (which will have a negative value) and the time taken could be calculated.
• This has a particular relevance to road safety. Highway Code braking distances can be used to calculate the time of journey during braking and the acceleration. Speeds may have to be converted to appropriate units for some students.