Final answer:
The racer's final velocity is 13.15 m/s. The racer saved 21.20 s by accelerating. The winner finishes 4.42 m ahead of the other racer.
Step-by-step explanation:
(a) To find the final velocity, we can use the equation:
v = u + at
Where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time. Plugging in the given values:
v = 10.0 m/s + (0.450 m/s²)(7.00 s)
Simplifying gives:
v = 10.0 m/s + 3.15 m/s
v = 13.15 m/s
The racer's final velocity is 13.15 m/s.
(b) To calculate the time saved, we need to find the time it would have taken to cover the remaining distance of 300 m at the racer's initial velocity of 10.0 m/s. Using the equation:
s = ut + 0.5at²
Where s is the distance, u is the initial velocity, a is the acceleration, and t is the time. Plugging in the values:
300 m = (10.0 m/s)t + 0.5(0.450 m/s²)(t²)
This is a quadratic equation that can be solved to find the time. The positive root of the equation is approximately 28.20 s. Therefore, the racer saved 28.20 s - 7.00 s = 21.20 s by accelerating.
(c) The winner's final position can be determined by finding the distance traveled during acceleration and the distance traveled at constant velocity. The distance traveled during acceleration can be found using the equation:
s = ut + 0.5at²
where s is the distance, u is the initial velocity, a is the acceleration, and t is the time. Plugging in the values:
s = (10.0 m/s)(7.00 s) + 0.5(0.450 m/s²)(7.00 s)²
This gives the distance traveled during acceleration as 295.58 m. The distance traveled at constant velocity is 300 m - 295.58 m = 4.42 m. Therefore, the winner finishes 4.42 m ahead of the other racer.