There are a lot of questions which involve two or more
people or cars travelling at various speeds along a road or path. Usually you
need to plot a *distance-time* graph for these types of questions. Then,
using the graph, you can find out interesting information. For instance, the
point on a graph where two lines cross over each other represents the time when
two cars or people were at the same location. Here’s an example question:

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Burke and Wills were two famous Australian explorers who
did many fantastic journeys into inner Australia. On 21 · Write two equations for the distance travelled from the camp by both groups ‘t’ hours after 5 pm. What is the ‘unknown’? · On the same set of axes plot graphs showing the pursuit, using ‘t’ as the x-axis and distance from the camp as the y-axis. Use the graph to work out whether Burke and Wills would have caught up in time. · Use the equations to get the answer as well. If they catch up, how far are they from the camp? |

Solution |

The equation for Burke and Will’s distance from the camp after 5 pm is easy, it’s just: The extra The equation for the backup party’s distance from base camp has to represent the fact that they’ve been travelling for 7 hours before 5 pm. In 7 hours they have travelled 17.5 kms, so we need a “+ 17.5 km” part in our equation: Lots of questions ask you what the unknown is. Think
about the question, and what you’re trying to find out. For instance, in
this question we’re interested in whether Burke and Wills will catch the
backup party. We have these variables – d The plot for the graphs is easy, they’re just two straight lines. The Burke and Will’s plot will start from the origin, but the backup party’s line will start some way up the distance axis to represent the fact they’re already 17.5 km from the camp at 5 pm. Now, how far ahead in time do we need to plot the graph? Well, Burke and Will’s time is up after 8 hours, so our time axis should only go for 8 hours: Looking at the graph, the lines cross over each other
around the 7 hour mark. This cross over point represents the point in time
when both groups are the Now let’s solve this problem using the equations. We
know that if B & W could continue at their speed for ever, they would
definitely catch the backup party, since they’re travelling faster than
them. We want to know however if it would take more or less than 8 hrs, since
if it would take more, then they wouldn’t catch them before collapsing from
exhaustion. The catch-up point is the point when For the catch-up point: This means we can Now we have one unknown value – the time ‘t’. We’ve got
rid of the other two unknowns, Great, we get the same answer as we did using our graphs. Burke and Wills will catch up with the backup party 7 hours after 5 pm, or at midnight. Don’t forget the very last bit of the question – working out how far from the camp they all are. Just use one of the distance equations and substitute this value of ‘t’ into them: |