Question

A tortoise and a hare engage in a race. A tortoise can run with a speed of 0. 15 m/s. A hare can run 25 times as fast as the tortoise. In the race, they start at the same time; however, the hare being very proud of his much higher speed stops for a nap for 5. 0 minutes. The tortoise wins the race by 35 cm. (a) what is the length of the race? (b) how long does the race take?.

Answer 1

Let D be the total length of the race, and T the time it takes for the tortoise to finish it. Then

D = (0.15 m/s) T

Let t be the time after the start of the race when the hare decides to take a nap. In this time, the tortoise travels a distance

(0.15 m/s) t

and the hare travels

25 (0.15 m/s) t = (3.75 m/s) t

For the next 5.0 min = 300 s, the tortoise keeps moving and covers a distance of

(0.15 m/s) (300 s) = 45 m

while the hare covers no distance.

After these 300 seconds, the remaining time for the tortoise to complete the race is (T - t - 300 s). In this same time, the hare loses the race by 35 cm = 0.35 m, so overall it has covered a total distance of

(3.75 m/s) t + 0 (300 s) + (3.75 m/s) (T - t - 300 s) = D - 0.35 m

Solve for T :

(3.75 m/s) t + 0 + (3.75 m/s) T - (3.75 m/s) t - 1125 m = D - 0.35 m

(3.75  m/s) T - 1125 m = D - 0.35 m

(3.75 m/s) T - 1125 m = (0.15 m/s) T - 0.35 m

(3.60 m/s) T = 1124.65 m

T = 312.403 s

(b) So, the race takes about 312 seconds, or 5.21 minutes.

(a) It also follows that the race has a total length of

D = (0.15 m/s) (312.403 s) ≈ 46.9 m