A bicycle racer sprints near the end of a race to clinch a victory. The racer has an initial velocity of 10.3 m/s and accelerates at the rate of 0.370 m/s2 for 7.01 s. What is his final velocity?
a) 12.9 m/s
The racer continues at this velocity to the finish line. If he was 237 m from the finish line when he started to accelerate, how much time did he save by accelerating?
b) 3.92 s
One other racer was 5.50 m ahead when the winner started to accelerate, but he was unable to accelerate, and traveled at a constant speed of 11.8 m/s until the finish line. How far behind (in meters) is he from the winner when the winner crosses the finish line?
How many more seconds did the other racer takes to cross the finish line?
a. V=Vo + a*t=10.3 + 0.37*7.01=12.9 m/s.
To find the final velocity of the racer, we can use the formula:
final velocity = initial velocity + (acceleration × time)
Given:
- Initial velocity (u) = 10.3 m/s
- Acceleration (a) = 0.370 m/s^2
- Time (t) = 7.01 s
Using the formula, we can calculate the final velocity:
final velocity = 10.3 m/s + (0.370 m/s^2 × 7.01 s)
final velocity = 10.3 m/s + 2.5927 m/s
final velocity ≈ 12.8927 m/s
So the final velocity of the racer is approximately 12.8927 m/s. Hence, option a) 12.9 m/s is correct.
To determine how much time the racer saved by accelerating, we can calculate the time it would have taken if he had not accelerated.
Distance remaining after acceleration = Total distance - Distance covered during acceleration
Distance remaining after acceleration = 237 m - (0.5 × acceleration × time^2)
Distance remaining after acceleration = 237 m - (0.5 × 0.370 m/s^2 × (7.01 s)^2)
Distance remaining after acceleration ≈ 237 m - 29.1794 m
Distance remaining after acceleration ≈ 207.8206 m
To find the time taken to cover the remaining distance at the constant velocity, we can use the formula:
time = distance / velocity
Given:
- Distance remaining after acceleration = 207.8206 m
- Constant velocity = 12.8927 m/s (final velocity from the previous calculation)
time = 207.8206 m / 12.8927 m/s
time ≈ 16.104 s
The time saved by accelerating is the difference between the total time taken and the time taken without acceleration:
time saved = total time - time without acceleration
time saved = 7.01 s - 16.104 s
time saved ≈ -9.094 s
We can discard the negative sign and consider the magnitude, so the time saved is approximately 9.094 s. Hence, option b) 3.92 s is incorrect.
To find the distance between the winner and the other racer when the winner crosses the finish line, we need to calculate the distances covered by both racers.
Distance covered by the winner = Distance covered during acceleration + Distance covered after acceleration
Distance covered by the winner = 0.5 × acceleration × (time during acceleration)^2 + final velocity × time after acceleration
Distance covered by the winner = 0.5 × 0.370 m/s^2 × (7.01 s)^2 + 12.8927 m/s × (16.104 s - 7.01 s)
Distance covered by the winner ≈ 72.9554 m + 140.1479 m
Distance covered by the winner ≈ 213.1033 m
Distance covered by the racer without acceleration = Constant velocity × time
Distance covered by the racer without acceleration = 11.8 m/s × (16.104 s - 7.01 s)
Distance covered by the racer without acceleration ≈ 111.824 m
Hence, the other racer is 213.1033 m - 111.824 m = 101.2793 m behind the winner when the winner crosses the finish line.
To find the additional time taken by the other racer to cross the finish line, we can use the formula:
time = distance / velocity
Given:
- Distance covered by the racer without acceleration = 101.2793 m
- Constant velocity = 11.8 m/s
time = 101.2793 m / 11.8 m/s
time ≈ 8.59 s
Therefore, the other racer takes an additional 8.59 s to cross the finish line.