A cyclist accelerates steadily from rest on a 500 m downhill stretch of road, reaching a speed of 22.0 m s^-1 before free-wheeling to a halt 192 s after starting. Calculate how long the cyclist takes to reach top speed.

Question

compphysgeek · Answer

do you have any information about the slope of the road after the first 500m?

tukitw · Answer

No

compphysgeek · Answer

If that's the only information that's given I have to assume that $v = 22 ms^{-1}$ the maximum speed is. the acceleration was constant over $s = 500 m$. 
The two equations we know for constant acceleration $a$ is $$ v = a t \ s = \frac{1}{2}a t^2$$
we want to know the time it took to reach the maximum speed and we don't know the acceleration.  So we solve the first equation for a and enter a into the second equation.$$ a = \frac{v}{t} \ s = \frac{v}{2t}t^2  = \frac{vt}{2}$$ We solve the equation for t $$t = \frac{2s}{v} $$
Enter the known values into the final equation and you get the time it takes to accelerate to $22 ms^{-1}$ over 500 m.

Without any other information there is nothing else to calculate here. If the slope of the road is downwards after the 500 m the cyclist would still be accelerating and he would never stop. If the slope was horizontal or upwards the maximum speed would be 22 m/s. Dunno what the information of 192 seconds is there.

tukitw · Answer

Thank you, it's the correct answer. I was confused about the free-wheeling part.