A bicycle racer sprints at the end of a race to clinch a victory. The racer has an initial velocity of 10.0 m/s and accelerates at the rate of 0.600 m/s2 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 10.2 m/s until the finish line. How far ahead of him (in meters and in seconds) did the winner finish?
distance

time

To find the answers to these questions, we can use the following kinematic equations:

1. Final velocity equation:
vf = vi + at

2. Distance traveled equation:
d = vit + (1/2)at^2

(a) To find the final velocity of the racer, we can use the first equation.
Given:
Initial velocity (vi) = 10.0 m/s
Acceleration (a) = 0.600 m/s^2
Time (t) = 7.00 s

Using the equation vf = vi + at, we substitute the given values:
vf = 10.0 m/s + (0.600 m/s^2)(7.00 s)
vf = 10.0 m/s + 4.2 m/s
vf = 14.2 m/s

Therefore, the final velocity of the racer is 14.2 m/s.

(b) To find the time saved by the racer, we need to calculate the time it takes for the racer to travel 300 m at a constant velocity.
Given:
Distance (d) = 300 m
Velocity (v) = 14.2 m/s

Using the equation d = vt, we rearrange it to solve for time (t):
t = d / v
t = 300 m / 14.2 m/s
t ≈ 21.13 s

Since the racer accelerated for 7.00 s before reaching the velocity of 14.2 m/s, we can subtract this time from the total time to find the time saved:
Time saved = total time - time spent accelerating
Time saved = 21.13 s - 7.00 s
Time saved ≈ 14.13 s

Therefore, the racer saved approximately 14.13 seconds.

(c) To find the distance and time by which the winner finished ahead of the other racer, we need to determine how long it took the winner to reach the finish line and then calculate the distance covered during that time.

Given:
Initial distance between racers = 5.00 m
Winner's acceleration = 0.600 m/s^2
Winner's initial velocity = 10.0 m/s
Opponent's constant velocity = 10.2 m/s

To find the time taken by the winner to reach the finish line, we can use the formula (vf = vi + at) with vf = 10.2 m/s and a = 0 (since the opponent did not accelerate):
Time taken = (vf - vi) / a
Time taken = (10.2 m/s - 10.0 m/s) / 0
Time taken = infinite

Since the time taken is infinite, it means that the winner never catches up to the opponent. Therefore, the opponent finishes the race ahead by 5.00 m.

In conclusion:
(a) The final velocity of the racer is 14.2 m/s.
(b) The racer saved approximately 14.13 seconds.
(c) The opponent finishes the race ahead of the winner by 5.00 meters.