Question

Susan and Katie are running an ultra marathon. Susan averages 5 miles per hour. Katie averages 5.2

miles per hour. If Susan gets a 45 minute head start, then exactly how many hours after Katie begins
running will she catch up to Susan?

Answer 1

Katie will catch up to Susan after running 33.4 h.

Explanation:

We can find the time that takes Katie to catch up to Susan by using the following equation:

`x_{f_(k)} = x_{0_(k)} + v_{0_(k)}t + (1)/(2)at^(2)`

Where:

`x_{f_(k)}`: is the final position of Katie

`x_{0_(k)}`: is the initial position of Katie = 0

`v_{0_(k)}`: is the initial speed of Katie = 5.2 mi/h

a: is the acceleration = 0 (she is moving at constant speed)

t: is the time

Since Katie will catch up to Susan, the final distance traveled by Katie will be equal to the final distance traveled by Susan.

`x_{f_(s)} = x_{0_(s)} + v_{0_(s)}t + (1)/(2)at^(2)`

`x_{0_(k)} + v_{0_(k)}t = x_{0_(s)} + v_{0_(s)}t`

Since Susan gets a 45 minutes head start, in that time she traveled the following distance:

`d = (v)/(t) = (5 mi/h)/(45 min*(1 h)/(60 min)) = 6.67 mi`

So, this will be the initial position of Susan.

`0 + v_{0_(k)}t = x_{0_(s)} + v_{0_(s)}t`

Hence, the time will be:

`5.2t = 6.67 + 5t`

`t = 33.4 h`

Therefore, Katie will catch up to Susan after running 33.4 h.

I hope it helps you!