Question

A Thomson's gazelle can run at very high speeds, but its acceleration is relatively modest. A reasonable model for the sprint of a gazelle assumes an acceleration of 4.2m/s2
for 6.5 s, after which the gazelle continues continues at a steady speed.
a. What is the gazelle's top speed?
b. A human would win a very short race with a gazelle. The best time for a 30 m sprint for a human runner is 3.6 s. How much time would the gazelle take for a 30 m race?
c. A gazelle would win a longer race. The best time for a 200 m sprint for a human runner is 19.3s.
How much time would the gazelle take for a 200 m race?

Answer 1

Answer:

a) 27.3 m/s

b) 3.78 s

c) 10.576 s

Explanation:

t = Time taken

u = Initial velocity

v = Final velocity

s = Displacement

a = Acceleration

a)

$v=u+at\\\Rightarrow v=0+4.2\times 6.5\\\Rightarrow v=27.3\ m/s$

Top speed of the gazelle is 27.3 m/s

b)

$s=ut+\frac{1}{2}at^2\\\Rightarrow 30=0t+\frac{1}{2}\times 4.2\times t^2\\\Rightarrow t=\sqrt{\frac{30\times 2}{4.2}}\\\Rightarrow t=3.78\ s$

The gazelle would take 3.78 seconds to win a 30 m race.

c)

$s=ut+\frac{1}{2}at^2\\\Rightarrow s=0\times 6.5+\frac{1}{2}\times 4.2\times 6.5^2\\\Rightarrow s=88.725\ m$

The gazelle would go 88.725 m with the acceleration after which there will be no acceleration.

Time = Distance / Speed

$\text{Time}=\frac{200-88.725}{27.3}=4.076\ s$

Total time taken by the gazelle to cover 200 m would be 6.5+4.076 = 10.576 seconds