In a quarter-mile drag race, two cars start simultaneously from rest, and each accelerates at a constant rate until it either reaches its maximum speed or crosses the finish line. Car A has an acceleration of 11.0 m/s2 and a maximum speed of 110 m/s. Car B has an acceleration of 11.8 m/s2 and a maximum speed of 91.8 m/s. (a) Which car wins the race? (b) By how many seconds does this car win the race?
To determine which car wins the race and by how many seconds, we need to calculate the time it takes for each car to cross the finish line.
(a) To find out which car wins, we need to compare the time it takes for each car to reach the finish line. The formula that relates acceleration, final velocity, and time is:
v = u + at
Where:
v = final velocity
u = initial velocity (in this case, 0 m/s since both cars start from rest)
a = acceleration
t = time
For Car A:
a = 11.0 m/s^2
v = 110 m/s
u = 0 m/s
Using the formula, we can rearrange it to solve for time:
t = (v - u) / a
t = (110 m/s - 0 m/s) / 11.0 m/s^2
t = 10.0 s
For Car B:
a = 11.8 m/s^2
v = 91.8 m/s
u = 0 m/s
Again, using the formula to solve for time:
t = (v - u) / a
t = (91.8 m/s - 0 m/s) / 11.8 m/s^2
t ≈ 7.78 s
Since Car B takes less time (7.78 s) to reach the finish line compared to Car A (10.0 s), Car B wins the race.
(b) To calculate the time difference between the two cars, we subtract the time for Car B from the time for Car A:
Time difference = Time for Car A - Time for Car B
Time difference = 10.0 s - 7.78 s
Time difference ≈ 2.22 s
So, Car B wins the race by approximately 2.22 seconds.
To determine which car wins the race and by how many seconds, we need to calculate the time it takes for each car to reach the finish line. We can use the equation of motion:
v^2 = u^2 + 2as
Where:
v = final velocity
u = initial velocity (which is 0 in this case as both cars start from rest)
a = acceleration
s = distance
For Car A:
The acceleration of Car A is 11.0 m/s^2, and the maximum speed is 110 m/s. The distance it needs to cover is a quarter-mile, which is equal to 402.336 meters.
To find the time taken by Car A, we first calculate the final velocity using the maximum speed:
v = 110 m/s
Next, we rearrange the equation of motion to solve for time (t):
v^2 = u^2 + 2as
110^2 = 0 + 2 * 11 * s
12100 = 22s
s = 550
Now we can find time (t):
t = s / v
t = 550 / 110
t = 5 seconds
Therefore, Car A takes 5 seconds to complete the quarter-mile race.
For Car B:
The acceleration of Car B is 11.8 m/s^2, and the maximum speed is 91.8 m/s. Using the same distance of 402.336 meters:
v = 91.8 m/s
We follow the same steps:
91.8^2 = 0 + 2 * 11.8 * s
8424.24 = 23.6s
s = 357.05
t = s / v
t = 357.05 / 91.8
t ≈ 3.89 (approximately)
Therefore, Car B takes approximately 3.89 seconds to complete the quarter-mile race.
Comparing both cars:
Car B, with a time of approximately 3.89 seconds, wins the race, while Car A takes 5 seconds to complete the race.
The difference in time is calculated by subtracting the time taken by Car B from the time taken by Car A:
Difference in time = 5 seconds - 3.89 seconds = 1.11 seconds
Therefore, Car B wins the race by 1.11 seconds.
Car A: 0.5a*Ta^2 = 400 m.
0.5*11*t^2 = 400. Ta = ?.
Car B: 0.5*11.8*Tb^2 = 400. Tb = ?.
b. T = Ta-Tb.