In the Daytona 500 auto race, a Ford Thunderbird and a Mercedes Benz are moving side by side down a straight-away at 67.5 m/s. The driver of the Thunderbird realizes he must make a pit stop, and he smoothly slows to a stop over a distance of 250 m. He spends 5.00 s in the pit and then accelerates out, reaching his previous speed of 67.5 m/s after a distance of 380 m. At this point, how far has the Thunderbird fallen behind the Mercedes Benz, which has continued at a constant speed?
To find out how far the Thunderbird has fallen behind the Mercedes Benz, we need to calculate the distance traveled by the Mercedes in the time it takes for the Thunderbird to come to a stop, spend time in the pit stop, and accelerate back up to its previous speed.
First, let's find the time it takes for the Thunderbird to come to a stop. We can use the equation:
v² = u² + 2as
where v is the final velocity (0 m/s since the Thunderbird comes to a stop), u is the initial velocity (67.5 m/s), a is the acceleration (which we'll need to calculate), and s is the distance (250 m).
Rearranging the equation to solve for a, we get:
a = (v² - u²) / (2s)
Substituting the values, we have:
a = (0² - 67.5²) / (2 * 250) = -227.43 m/s²
Since the acceleration is negative, it means the Thunderbird is slowing down.
Next, we can calculate the time it takes for the Thunderbird to decelerate from 67.5 m/s to 0 m/s using the equation:
v = u + at
where v is the final velocity (0 m/s), u is the initial velocity (67.5 m/s), a is the acceleration (-227.43 m/s²), and t is the time.
Rearranging the equation to solve for t, we get:
t = (v - u) / a
Substituting the values, we have:
t = (0 - 67.5) / -227.43 = 0.297 s
So, it takes approximately 0.297 seconds for the Thunderbird to come to a stop.
Next, the Thunderbird spends 5.00 seconds in the pit stop. Therefore, the total time it takes for the Thunderbird to come to a stop, spend time in the pit stop, and accelerate back up to its previous speed is:
Total time = 0.297 s (deceleration time) + 5.00 s (pit stop time) + x (acceleration time)
To find x, we can rearrange the acceleration equation:
v = u + at
to solve for t:
t = (v - u) / a
For the Thunderbird to reach 67.5 m/s, the final velocity (v) is 67.5 m/s, the initial velocity (u) is 0 m/s, and the acceleration (a) is the same as the deceleration (-227.43 m/s²).
Substituting the values, we have:
t = (67.5 - 0) / -227.43 ≈ -0.296 s
Since time cannot be negative, we ignore the negative sign and take the absolute value, so the Thunderbird takes approximately 0.296 seconds to accelerate back to 67.5 m/s.
Now, we can calculate the distance traveled by the Mercedes in this total time:
Distance = speed × time
Distance = 67.5 m/s × (0.297 s + 5.00 s + 0.296 s)
Distance ≈ 67.5 m/s × 5.593 s ≈ 377.47 m
Therefore, at this point, the Thunderbird has fallen behind the Mercedes Benz by approximately 377.47 meters.