# College Physics by Openstax| Problem 2.38

A bicycle racer sprints at the end of a race to clinch a victory. The racer has an initial velocity of 11.5 m/s and accelerates at the rate of $0.500 m/s^2$ for 7.00 s.

(a) What is his final velocity?

(b) The racer continues at this velocity to the finish line. If he was 300 m from the finish line when he started to accelerate, how much time did he save?

(c) One other racer was 5.00 m ahead when the winner started to accelerate, but he was unable to accelerate, and traveled at 11.8 m/s until the finish line. How far ahead of him (in meters and in seconds) did the winner finish?

SOLUTION:

Part a $v=v_o+at$ $v=11.5\:m/s+\left(0.500\:m/s^2\right)\left(7.00\:s\right)$ $v=15.0\:m/s$

Part b

Let $t_{const}$ be the time it takes to reach the finish line without accelerating: $t_{const}=\frac{x}{v_o}$ $t_{const}=\frac{300\:m}{11.5\:m/s}$ $t_{const}=26.09\:s$

Now let be the distance traveled during the 7 seconds of acceleration. We know t=7.00 s so $d=v_ot+\frac{1}{2}at^2$ $d=\left(11.5\:m/s\right)\left(7.00\:s\right)+\frac{1}{2}\left(0.500\:m/s^2\right)\left(7.00\:s\right)^2$ $d=92.75\:m$

Let t’ be the time it will take the rider at the constant final velocity to complete the race: $t'=\frac{x-d}{v}$ $t'=\frac{300\:m-92.75\:m}{15.0\:m/s}$ $t'=13.82\:s$

So the total time it will take the accelerating rider to reach the finish line is $T=t+t'$ $T=\:7\:s+13.82\:s$ $T=20.82\:s$

Finally, let T* be the time saved. So $T*=26.09\:s-20.82\:s$ $T*=5.27\:s$

Part c

Let $t_2$ be the time it takes for rider 2 to reach the finish line. $t_2=\frac{x'}{v_o'}$ $t_2=\frac{295\:m}{11.8\:m/s}$ $t_2=25.0\:s$

The time difference is $=t_2-T$ $=25.0\:s-20.817\:s$ $=4.2\:s$

Therefore, he finishes 4.2 s after the winner.

When the other racer reaches the finish line, the winner has been traveling at 15 m/s for 4.2 seconds, so the other racer finishes $x=\left(4.2\:s\right)\left(15\:m/s\right)$ $x=63\:m$

behind the other racer.

You can now buy the complete solution manual of College Physics by Openstax. Just click on the button you can see at the right portion of this post.

Advertisements