Using the time recorded to run 45 m, how many metres head start does the slower of two runners need in order for both runners to complete a 100 m race at the same time?

Question

faster runner = 7.98 seconds /45m

slower runner = 11.74 seconds / 45m

how many metres head start does the slower of two runners need in order for both runners to complete a 100 m race at the same time?

and if the runners were to start at opposite ends of the 100 m running track, where and when would they cross paths?

Answer 1

Firstly, I would find the velocities of each runner with the given times and distances.

Fast Runner= 45m/7.98 seconds= 5.64 meters/second

Slow Runner= 45m/ 11.74 seconds= 3.83 meters/second

If both runners need to cross the 100m line at the same time, then we know that the time it takes the fast runner to go 100 meters is the same time it should take the slower runner to go "x" meters. We can solve how long it would take the fast runner to go 100 meters.

time=100 meters/(5.64 m/s)= 17.73 seconds.

Now we know the time for both runners and we know the velocity of the slow runner, so we can find the distance for the slower runner.

x/3.83=17.73, x=67.91 meters. This means that the slow runner needs a 32.09 meter head start (100-67.91)

For the second part, we need to find an expression for both runners that connects time and distance. For the fast runner, that expression would be 5.64(t)=x. For the slower runner starting from the opposite side of the 100m track, it would be 3.83(t)=100-x. We can now solve two equations with two unknowns, giving us t=10.56 seconds and x=59.56 meters. (from the fast runner's side).