Question

BB8 is an intergalactic robot (essentially a soccer ball with a half meter radius that can control how fast it rolls around) that is trying out for a new role in the rebellion. One of the tests is how fast they can travel 100 m. They must start from rest and end at rest. BB8 has a maximum angular acceleration rate of 2 rad/s2, both when speeding up and slowing down. What is the minimum time BB8 can run the trial?

Answer 1

Answer:

20 seconds

Explanation:

t = Time taken

u = Initial velocity

v = Final velocity

s = Displacement

r = Radius of robot = 0.5 m

$\alpha$ = Angular acceleration = 2 rad/s²

Linear acceleration is given by

$a=r\alpha\\\Rightarrow a=0.5\times 2\\\Rightarrow a=1\ m/s^2$

$s=ut+\frac{1}{2}at^2\\\Rightarrow 50=0t+\frac{1}{2}\times 1\times t^2\\\Rightarrow t=\sqrt{\frac{50\times 2}{1}}\\\Rightarrow t=10\ s$

While speeding up time taken is 10 seconds

$v=u+at\\\Rightarrow v=0+1\times 10\\\Rightarrow v=10\ m/s$

$s=ut+\frac{1}{2}at^2\\\Rightarrow 50=10t-\frac{1}{2}\times 1\times t^2\\\Rightarrow t=\sqrt{\frac{50\times 2}{1}}\\\Rightarrow 50=10t-0.5t^2\\\Rightarrow -5t^2+100t-500=0$

$t=\dfrac{-100\pm \sqrt{100^2-4\left(-5\right)\left(-500\right)}}{2\left(-5\right)}\\\Rightarrow t=10\ s$

The time taken to slow down is 10 seconds

Total time the trial is rum is 10+10 = 20 seconds