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Sample problemset from the story The Tortoise and the Hare

Story: One day a hare was bragging about how fast he could run. He bragged and bragged and even laughed at the tortoise, who was so slow. The tortoise stretched out his long neck and challenged the hare to a race, which, of course, made the hare laugh.

"My, my, what a joke!" thought the hare.

"A race, indeed, a race. Oh! what fun! My, my! a race, of course, Mr. Tortoise, we shall race!" said the hare.

The forest animals met and mapped out the course. The race begun, and the hare, being such a swift runner, soon left the tortoise far behind. About halfway through the course, it occurred to the hare that he had plenty of time to beat the slow trodden tortoise.

"Oh, my!" thought the hare, "I have plenty of time to play in the meadow here."
 And so he did.

After the hare finished playing, he decided that he had time to take a little nap.
"I have plenty of time to beat that tortoise," he thought. And he cuddle up against a tree and dozed.

The tortoise, in the meantime, continued to plod on, albeit, it ever so slowly. He never stopped, but took one good step after another.

The hare finally woke from his nap. "Time to get going," he thought. And off he went faster than he had ever run before! He dashed as quickly as anyone ever could up to the finish line, where he met the tortoise, who was patiently awaiting his arrival.

Sample Problem

Problems:

In the hare and tortoise race, the tortoise runs with uniform speed of 10 cm/s and reaches the destination in 72 min. The hare runs at 1 m/s for the first 6 minutes and then sleeps for 65 minutes. He then runs at 1 m/s to reach the destination.

  1. What was the length of the race?
  2. When the hare went for sleep, what was the position of the tortoise? What was the position of the hare at this time?
  3. At what time did the tortoise cross over the hare?
  4. When the tortoise reached the destination, where was the hare?
  5. By how much time the hare lost the race?
  6. Plot the position versus time graph on the same plot for the hare and for the tortoise.
  7. In the situation described above there are several assumptions. Some of these assumptions are unphysical too. List all such assumptions and state if they are unphysical.

Solutions:

  1. Speed of tortoise (v) = 10 cm/s = 0.1 m/s
    Speed of hare (u) = 1 m/s
    Time taken by tortoise to win the race (t) = 72 mins.
    Total distance covered by the tortoise (d) = v x t = 0.1 x 72 x 60 = 432 m
    Hence, the length of the race was 432 m.
  2. Distance travelled by the hare in 6 minutes (dh) = 1 x 6 x 60 = 360 m
    Distance travelled by the tortoise in 6 minutes (dt) = 0.1 x 6 x 60 = 36 m
    When the hare went for sleep, the positions of hare and tortoise are (see Fig. 1):
    hare will be at point C (dh = 360 m from the origin)

    tortoise will beat point B(dt = 36 m from the origin)
  3. The time at which tortoise cross over the hare at point C is,
    tth = dt/v = 360/0.1 = 3600 s = 60 min.
  4. Time taken by the tortoise to reach the destination, t = 72 min.
    Time at which the hare wakes up = 6 + 65 = 71 min.
    Thus, hare had run for one min after waking up when the tortoise reached the destination.
    The distance travelled by the hare in one min. = 1x 1 x 60 = 60 m
    Hence, the hare was atpoint D (420 m from origin) when the tortoise reached the destination.
  5. The hare was 12 m away from the destination when the race was over.
    Thus, the hare lost the race by 12m/(1m/s) = 12 seconds

  6. Assumptions:
    1. The accelerations and declarations are infinity.
    2. Cross over time is neglected, i.e., the both, hare and tortoise, are considered to be of zero size.