Answer to Question #157478 in Physics for Alizah Khan

Question

1. The Cheetah A cheetah on the prowl is stalking an antelope on the African plains. The cheetah slowly creeps forward towards the antelope until the cheetah is only 35 meters away, at which point it springs out from the brush and accelerates towards the antelope! The cheetah accelerates at a constant rate for 3.0 seconds, at which point it reaches its top speed of 27 m/s. The cheetah can continue running at this top speed for another 8.0 seconds before it becomes tired and slows down to a stop. If the antelope wishes to escape unharmed, how fast should the antelope run away from the cheetah? Assume the antelope attains its maximum speed in a negligible amount of time, and that the antelope can run at its maximum speed for much longer than the cheetah can.

Answer 1

Calculate the distance passed by the cheetah:

$$d=\frac{v_ft}{2}=40.5\text{ m}.$$

It took the cheetah 3 seconds to pass this distance, while the antelope is 35 m away.

Assume that the antelope is a bit lazy, tired, or risky, and it doesn't feel too anxious if the cheetah is rushing right behind. Thus, the antelope starts running at a speed that keeps it 40.5-35=5.5 m away from the point where the antelope was initially when the cheetah approaches.

Calculate the time required for the cheetah to cover the distance between d=35 and 40.5 m:

$$a=\frac{v_f}{t}=9\text{ m/s}^2,\\\space\\ t_1=\sqrt{\frac{2d}{a}}=2.7889\text{ s},\\\space\\ t_2=t-t_1=3-2.7889=0.2111\text{ s}.$$

So, the antelope must cover 5.5 m in 0.2111 s, which corresponds to a velocity of

$$u=\frac{x}{t_2}=\frac{5.5}{0.2111}=26.5\text{ m/s}.$$

Since the cheetah is chasing at 27 m/s, the antelope is at risk. However, if the antelope starts rushing at 27 m/s, it will save itself from the cheetah's paws.