Question

Races in athletics are started by the sound of a starting gun, traditionally located to one side of the
first lane. But are runners disadvantaged if they are further away from the person holding the
starting gun?
Task
Using algebra, investigate whether the distance from the starting gun will impact on a runner’s ability
to win a race. Explore which runners would be most disadvantaged if the race was won by 0.1 seconds
or 0.01 seconds.
Useful information
 Runners are located side by side in a straight line to start the race.
 Runners are placed 1 metre apart. The furthest runner is 11 m from the starting gun, the
closest is 1 metre.
 Speed of sound is approximately 340 metres per second.
Steps to help you solve the problem
 Develop a formula that relates speed of the starting gun sound to distance of the runner from
the starting gun and the time the sound takes to travel this distance.
 Rewrite the formula with time as the subject.
 Use the formula to determine the times taken for sound to reach each of the runners.
 Compare this to the finishing times proposed of the winner.

Answer 1

First let us chalk down the given conditions :

1. Runners are located side by side in a straight line to start the race

2. Runners are placed 1 metre apart

3. The furthest runner is 11m from the starting gun and the closest one is 1 metre apart from the starting gun

4. Speed of sound is approximately 340m/s

So, we see by implication rule that :

340metres 1second

1metre →? seconds

Whhich gives the time taken by the sound of the starting gun to reach the first runner to be :

340 second

So, we develop a formula to find the time taken for that sound of the starting gun to reach the subsequent runner who are spaced one metre apart to be :

340seconds
where, is the position of the runner from the starting gun

This is developed from the instantaneous speed formula :

Speed = x/Time

where x is the distance from the starting gun

To express this formula in terms of time, we use,

Time-Speed

$\Rightarrow Time = \frac{x}{340}$

Time taken by the sound to reach each of the 11 runners is :

$t_{1} = \frac{1}{340} = 0.0029seconds$

0.0059seconds 2340

$t_{3} = \frac{3}{340} = 0.0088seconds$

$t_{4} = \frac{4}{340} = 0.0118seconds$

$t_{5} = \frac{5}{340} = 0.0147seconds$

$t_{6} = \frac{6}{340} = 0.0176seconds$

$t_{7} = \frac{7}{340} = 0.0206seconds$

$t_{8} = \frac{8}{340} = 0.0235seconds$

$t_{9} = \frac{9}{340} = 0.0265seconds$

$t_{10} = \frac{10}{340} = 0.0294seconds$

$t_{11} = \frac{11}{340} = 0.0324seconds$

(iv)

So, if the first runner won the race and the 11th runner came second, he would be at a disadvantage if the time difference between the two is around 0.01seconds because,

So, if 1st runner came first the following runners would be at a loss if they came second :

So if runner 1 came first, the runners 5 to 11 will be at a loss if they came second by a difference of 0.01seconds

Similarly, for runner 2 coming first, the runners 6 to 11 will be at a loss if they came second by a difference of 0.01 seconds

If runner 3 came first, the runners 7 to 11 will be at a loss if they came second by a difference of 0.01 seconds

If runner 4 came first, the runners 8 to 11 will be at a loss if they came second by a difference of 0.01 seconds

If runner 5 came first, the runners 9 to 11 and runner 1 will be at a loss if they came second by a difference of 0.01 seconds

If runner 6 came first, the runners 10, 11, 1 & 2 will be at a loss if they came second by a difference of 0.01 seconds

If runner 7 came first, the runners 1, 2, 3 & 11 will be at a loss if they came second by a difference of 0.01 seconds

If runner 8 came first, the runners 1 to 4 will be at a loss if they came second by a difference of 0.01 seconds

If runner 9 came first, the runners 1 to 5 will be at a loss if they came second by a difference of 0.01 seconds

If runner 10 came first, the runners 1 to 6 will be at a loss if they came second by a difference of 0.01 seconds

If runner 11 came first, the runners 1 to 7 will be at a loss if they came second by a difference of 0.01 seconds

So, it can be concluded that the distance of the starting gun can make an impact on the runner's ability to win the race.