In the Daytona 500 auto race, a Ford Thunderbird and a Mercedes Benz are moving side by side down a straight-away at 66.5 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 66.5 m/s after a distance of 310 m. At this point how far has the Thunderbird fallen behind the Mercedes Benz, which has continued at a constant speed?
time to slow: distance/avg velocity= 250/(66.5/2) seconds
time to accelerate: distance/avg velocity= 310/(66.5/2)seconds.
timelost= time to slow+timeinpit+time accelerating
distance front car went: 66.5*timelost
distance pit car went: 250+310
the difference in those is how far they are apart.
560 m
To calculate the distance by which the Thunderbird has fallen behind the Mercedes Benz, we need to find the distance covered by each vehicle during the time the Thunderbird spends in the pit stop.
First, let's find the time it takes for the Thunderbird to decelerate to a stop and accelerate back up to its previous speed.
Initial speed of the Thunderbird (v1) = 66.5 m/s
Final speed of the Thunderbird (v2) = 0 m/s (when coming to a stop)
Acceleration of the Thunderbird (a) = (v2 - v1) / t
= (0 - 66.5) / t
= -66.5 / t
Here, t represents the time taken by the Thunderbird to decelerate and accelerate.
Using the formula for distance covered during acceleration or deceleration:
Distance (d) = (v1 + v2) / 2 * t
250 m = (66.5 + 0) / 2 * t
250 m = 33.25 * t
t = 250 / 33.25
t ≈ 7.52 s
Now, let's find out how far the Thunderbird would have traveled during this time at a constant speed of 66.5 m/s:
Distance covered during 7.52 seconds at 66.5 m/s = 7.52 * 66.5
≈ 499.18 m
Therefore, the Thunderbird has fallen behind the Mercedes Benz by a distance of approximately 499.18 meters.
To find out how far the Thunderbird has fallen behind the Mercedes Benz, we need to calculate the distance traveled by the Mercedes Benz during the time the Thunderbird was in the pit stop.
First, let's calculate the time it took for the Thunderbird to slow down and stop, and its subsequent acceleration to reach its previous speed.
The Thunderbird slows down from 66.5 m/s to a stop over a distance of 250 m. We can use the equation of motion:
v^2 = u^2 + 2as
Where:
v = final velocity (0 m/s since the Thunderbird comes to a stop)
u = initial velocity (66.5 m/s)
a = acceleration (unknown)
s = distance (250 m)
Rearranging the equation:
a = (v^2 - u^2) / (2s)
a = (0^2 - 66.5^2) / (2 * 250)
a = -22212.25 / 500
a = -44.42 m/s^2
Now, let's find the time it took for the Thunderbird to slow down and come to a stop. We can use the equation of motion:
v = u + at
Where:
v = final velocity (0 m/s)
u = initial velocity (66.5 m/s)
a = acceleration (-44.42 m/s^2, since it is decelerating)
t = time (unknown)
Rearranging the equation:
t = (v - u) / a
t = (0 - 66.5) / -44.42
t = 1.5 seconds
So, it takes the Thunderbird 1.5 seconds to slow down and come to a stop.
Next, we need to find the time it takes for the Thunderbird to accelerate from 0 m/s to 66.5 m/s.
Using the equation of motion:
v = u + at
Where:
v = final velocity (66.5 m/s)
u = initial velocity (0 m/s)
a = acceleration (unknown)
t = time (unknown)
Rearranging the equation:
t = (v - u) / a
t = (66.5 - 0) / a
Since the acceleration is the same for both scenarios (deceleration and acceleration), we can use the same acceleration value calculated earlier, which is -44.42 m/s^2.
t = (66.5 - 0) / -44.42
t = -1.5 seconds
Note: The negative sign indicates that the Thunderbird is accelerating in the opposite direction of its initial motion.
So, it takes the Thunderbird 1.5 seconds to accelerate back to its previous speed.
Now, let's calculate the distance traveled by the Mercedes Benz while the Thunderbird was in the pit stop.
Distance = Speed × Time
The Mercedes Benz continues at a constant speed of 66.5 m/s throughout the Thunderbird's pit stop of 1.5 seconds.
So, the distance traveled by the Mercedes Benz during this time is:
Distance = 66.5 m/s × 1.5 s
Distance = 99.75 meters
Therefore, the Thunderbird has fallen behind the Mercedes Benz by a distance of 99.75 meters at the point when the Thunderbird reaches its previous speed of 66.5 m/s after the pit stop.