Having looked at the first Physics paper for this year’s AQA Trilogy examination, it seems to me that a significant proportion of questions have been framed around situations and applications rather than just recalling and connecting facts. With that in mind, let’s take a real-world look at motion along a line…
First an important definition. Movement along a line that changes direction is always about increasing the distance travelled whereas movement in a straight line is about a change in displacement, which can either increase and decrease.
You should recall that displacement is the vector form of distance: it is distance from a fixed point when measured in a straight line, at a stated angle. I have written about vectors and scalars before: it is extremely likely that there will be a question about vectors and scalars in Paper 2 so make sure you know both the definitions and some examples. Read this article to refresh your memory.
The diagram below shows the difference between distance and displacement in the context of a running track. We will consider a 200 m race: the lanes show the routes that runners must take whereas the straight lines show the displacement for the runner in Lane 1 at two points during the race. Note that the displacements are vectors so there are arrows on the lines to indicate the appropriate direction: when drawing vectors, arrows are essential.
The runners start at different locations to ensure that the distance travelled is the same (200 m) for all competitors. But the displacement of the runners from the finish is not the same at the start of the race. In fact, the displacement can increase during the race, even when the remaining distance is decreasing, as shown for Lane 1 in the diagram above. The magnitude (length) of displacement vector B is greater than the magnitude of displacement vector A even though the runner’s distance to the finish when rounding the curve is less than it was at the start. So distance and displacement are clearly very different things.
In fact, if you were to make exactly one lap of the track, the distance covered would be 400 m but your displacement, going from the start to the finish, would be zero. Why? Because you have finished right back where you started and the length of the vector connecting these two points is therefore zero. So the displacement is zero.
Taking a short cut across a field when walking home is another example of using displacement instead of walking the full distance around corners. You might spot that corners and curves are the features that make the distance travelled greater than the displacement between the start and the finish of a journey – and you would be quite right. The other way of looking at displacement is to say it is the shortest distance between two points (because it involves moving in a straight line, with no changes of direction).
You should be able to calculate the time taken to complete a journey using either distance or displacement information. In both cases, the same equation applies – but there is potential confusion here and this is what you need to know…
We usually say, “speed is distance divided by time”. That is true but when it comes to putting this expression into symbols, there are two forms of the equation and you need to recognise both of them, not least because you could be asked to explain what the symbols mean. Here are the two forms and a bit of explanation about each;
The above equation matches what we say. You have probably been using this version for five or six years so it should be very familar to you.
This is exactly the same equation but using velocity instead of speed. It also has displacement in place of distance – and displacement has the symbol s (which is confusing if you automatically think of s as the symbol that represents speed).
Things actually look worse if the second equation is rearranged to calculate displacement, because now we have s on the left of the equation – and the s doesn’t stand for speed!
Note that the multiplication symbol is normally dropped – as is the case in maths expressions when we write, for example, 2y. You know that means “two times y” but it would be very confusing to write 2 x y.
To finish, here are a few more examples of distance and displacement.
Think about Jorge practising penalty shots but on one occasion (just one) the basketball bounces off the top of the hoop and comes back to him. Describe the distance travelled by the ball. At what point does the ball reach its maximum displacement from Jorge and what is the ball’s total displacement?
Now think about a tennis warm-up. Two players are on opposite sides of the net and they are knocking the ball back and forth at a constant rate. With every hit of the ball, the ball covers more distance. It is important to realise that distance can never decrease: for all of the time when an object is moving, it is covering more distance.
But what about the tennis ball’s displacement? Assuming an idealised situation where the players mainatin a steady pace of hitting the ball to each other, the ball’s displacement will go through a cycle that keeps on repeating.
The graphs below show the displacement of the tennis ball but they are with respect to two different measurement points. One is the graph for measuring displacement relative to the net. When the ball is moving over the net in one direction it will have a positive displacement and when it is hit back in the opposite direction it will move into negative displacement. The other graph measures displacement relative to one of the players, when the displacement will simply increase then decrease. Can you identify which is which? For an extra callenge, can you describe where to place a horizontal line that marks the position of the net? (The position is different on the two graphs.)
CONFESSION! I originally intended to give graphs for a pendulum’s movement instead of the tennis ball example above but the shape for a pendulum is a sine curve (which looks more complicated) rather than sawtooth. Can you explain why the sawtooth shape would be wrong for a pendulum – you definitely do not need to know this for the GCSE course! HINT: Think about how the velocity of a pendulum changes as it swings from side to side (and how that is different from the velocity profile of a tennis ball being struck by rackets). The mathematics behind pendulums is a topic known as simple harmonic motion (SHM) which you’ll meet if you go on to study A-Level Physics.
Mr Tarrant's physbang 'blog 