A cyclist leaves his training base for a morning workout, riding at the rate of 18 mph. One hour later, his support staff leaves the base in a car going 52 mph in the same direction. How long will it take the support staff to catch up with the cyclist once the car leaves the base?
At the one hour point the cyclist is 18 miles from from the training base and his support staff.
From this point on, his support staff will be closing in on the cyclist at the closing speed of 52 - 18 = 34mph.
Thje initial gap of 18 miles will therefore be closed in 18/34 = .514285 hours or 30min - 51.42sec.
Since car #1 has a 1 hr head-start,
d1 = 18 mi/hr * 1hr + 18t,
d1 = 18 + 18t,
d2 = 52t,
When car #2 catches up:
d1 = d2,
18 + 18t = 52t,
18 = 52t - 18t = 34t,
t = 18 / 34 = 0.5294 hrs = 31.8 min
To find out how long it will take the support staff to catch up with the cyclist, we need to determine the distance the cyclist covers in the one hour before the car starts, and then calculate the time it takes for the car to cover that distance.
Let's start by finding the distance the cyclist covers in one hour. We can do this by multiplying the cyclist's speed, 18 mph, by the time of 1 hour:
Distance covered by the cyclist = Speed x Time
Distance covered by the cyclist = 18 mph x 1 hour
Distance covered by the cyclist = 18 miles
Since the cyclist covers a distance of 18 miles in one hour, the support staff needs to cover the same distance to catch up.
Now, we can calculate how long it takes for the car to cover 18 miles. We can use the formula for time:
Time = Distance / Speed
Time taken by the car = 18 miles / 52 mph
Time taken by the car ≈ 0.346 hours (rounded to three decimal places)
Therefore, it will take the support staff approximately 0.346 hours, or about 20.76 minutes, to catch up with the cyclist once the car leaves the base.