Thinking about journeys

The journey described by our graphs was along a straight road. This is a very simplistic and unlikely journey in the real world.

For each of the situations below, can you sketch a distance-time graph and a displacement-time graph?

  • Do the graphs look the same?

  • Are they both possible to draw?

Situation one

You walk straight to school travelling at a constant speed of \(\quantity{0.5}{m\,s^{-1}}\).

Distance between home and school is 150 metres.
Graph of distance is a straight line through the origin.
Graph of displacement is a straight line through the origin.

We didn’t state where we were measuring displacement from. The graph above shows the displacement measured from ‘home’. If instead displacement was measured from ‘school’, how would the displacement graph change?

  • Would the distance graph also change?
  • How many dimensions is our journey taking place in?

Situation two

You walk straight along the road from home to school and at the end of the first road you take a \(90^{\circ}\) turn left. You travel at a constant speed of \(\quantity{0.5}{m\,s^{-1}}\).

Walk 100 metres from home, turn left and walk 50 metres.

At faster speeds we would expect to slow down when we approach and turn a corner. Here we are making the assumption that the constant speed of \(\quantity{0.5}{m\,s^{-1}}\) is maintained.

The distance time graph is a straight line passing through the origin.

While the distance-time graph is easy to draw (and is the same as above), the displacement-time graph may have caused a few more problems. Remember, we defined displacement as how far you are from the starting point (assuming displacement is being measured from there) so, while you might be moving at \(\quantity{0.5}{m\,s^{-1}}\), you are not moving away from the starting point at this speed.

The journey diagram showing the displacement.

When the distance travelled is \(\quantity{125}{m}\), the displacement seems to be \[\sqrt{100^2+25^2}\approx \quantity{103}{m}.\]

Does our answer of \(103\) m make sense?

In Situation one, we were travelling in one dimension, so we had forward (positive displacement) and backward (negative displacement). If I told you that you were \(103\) m away from your starting point in a two dimensional world, would you be able to tell me where you were?

We need extra information. Displacement is a vector quantity which tells us both the distance from the starting point and the direction. In one dimension, the direction could be either positive or negative, but in two dimensions that is not enough.

We might give an angle along with our distance. For example, \(103\)m at an angle of \(14.04^{\circ}\), although we would have to know where our angle is being measured from and in which direction.

The displacement vector when we have walked 25 metres.

Or we might use our two dimensions as separate directions, at right angles to one another, that can be positive or negative. (This should sound familiar!) We could write the displacement as \(\begin{pmatrix} 100 \\ 25 \end{pmatrix}\). Where have you seen something like this before?

The journey and displacement diagram in a coordinate system.

Either way, this represents our displacement at a single point, and not the displacement throughout the journey. That displacement cannot meaningfully be sketched on a displacement-time graph.

Representing a simple journey in two dimensions can become complicated when we think about displacement, rather than distance travelled. Does the same happen with speed and velocity? These questions are considered in Speed vs velocity.