In the Daytona 500 auto race, a Ford Thunderbird and a Mercedes Benz are moving side by side down a straightaway at 78.0 m/s. The driver of the Thunderbird realizes that she must make a pit stop, and she smoothly slows to a stop over a distance of 250 m. She spends 5.00 s in the pit and then accelerates out, reaching her previous speed of 78.0 m/s after a distance of 290 m. At this point how far has the Thunderbird fallen behind the Mercedes Benz, which has continued at a constant speed?
m
To find out how far the Thunderbird has fallen behind the Mercedes Benz, we need to calculate the distance traveled by the Mercedes during the time the Thunderbird is in the pit stop.
First, let's calculate the time it takes for the Thunderbird to come to a stop and accelerate back to its original speed.
The initial speed of the Thunderbird is 78.0 m/s, and it comes to a stop over a distance of 250 m. We can use the formula for acceleration to find the time it takes:
v^2 = u^2 + 2as
0^2 = 78.0^2 + 2(a)(250)
0 = 6084 + 500a
-6084 = 500a
a = -6084/500
a = -12.168 m/s^2
Now, we have the acceleration value. Let's calculate the time it takes for the acceleration phase.
v = u + at
0 = 78.0 + (-12.168)t
12.168t = 78.0
t = 78.0/12.168
t ≈ 6.40 s
After spending 5.00 s in the pit stop, the Thunderbird accelerates back to its original speed. So the total time spent in the pit stop is:
Total time = time to decelerate + time in the pit stop = 6.40 s + 5.00 s = 11.40 s
Now, we can calculate the distance traveled by the Mercedes during this time.
The Mercedes is moving at a constant speed of 78.0 m/s. So we can use the equation:
distance = speed × time
distance = 78.0 m/s × 11.40 s
distance ≈ 889.2 m
Therefore, the Thunderbird has fallen behind the Mercedes by approximately 889.2 m.