In the Daytona 500 auto race, a Ford Thunderbird and a Mercedes Benz are moving side by side down a straightaway at 71.0 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 71.0 m/s after a distance of 300 m. At this point how far has the Thunderbird fallen behind the Mercedes Benz, which has continued at a constant speed?
v^2 = 2as
71^2 = 500a
a = -10 m/s^2
t1 = 71/10 = 7.1s to stop
Now work the same magic with s=300 to find the time t3 to accelerate, then add the three times together
The Benz has gone x = 71.0 (t1+5+t3) meters
so the T-bird is x-(250+300) meters behind
time to slow down:
Vi = 71.0 m/s
average speed during slowing = 71/2 = 35.5 m/s
250 = 35.5 t
t = 7.04 seconds to stop
T now at 250 meters
M now at 7.04*71 = 500 meters (twice as fast of course)
after 5 second stop
T still at 250 meters
M now at 500 + 5*71 = 855 meters
now T 300 meters at 35.5 m/s again so 300/35.5 = 8.45 s
T now at 250 + 300 = 550 meters
M now at 855 + 8.45*71 = 1455 meters
1455 - 550 = ?
Thanks so much!!!!!
To calculate how far the Thunderbird has fallen behind the Mercedes Benz, we need to find the distance traveled by both vehicles during the time the Thunderbird spends in the pit.
First, let's calculate the time it takes for the Thunderbird to slow down and stop. We know the initial speed (71.0 m/s) and the distance traveled during this deceleration (250 m). We can use the formula:
v^2 = u^2 + 2as
Where:
v = final velocity (0 m/s as the Thunderbird stops)
u = initial velocity (71.0 m/s)
a = acceleration (unknown)
s = distance (250 m)
Rearranging the formula to solve for acceleration:
a = (v^2 - u^2) / (2s)
Plugging in the values:
a = (0^2 - 71.0^2) / (2 * 250)
Calculating the acceleration (a):
a = -2548 m/s^2 (negative because it is deceleration)
Now we can use the acceleration (a) and distance (300 m) to find the time it takes for the Thunderbird to accelerate back to 71.0 m/s using the formula:
v = u + at
Where:
v = final velocity (71.0 m/s)
u = initial velocity (0 m/s as the Thunderbird starts from a stop)
a = acceleration (-2548 m/s^2)
t = time (unknown)
Rearranging the formula to solve for time:
t = (v - u) / a
Plugging in the values:
t = (71.0 - 0) / (-2548)
Calculating the time (t):
t ≈ -0.028 s (negative because the time is measured backward from the end point to the starting point)
Now, we can add the time spent in the pit (5.00 s) to the time required for acceleration (-0.028 s) to find the total time the Thunderbird is behind the Mercedes Benz:
Total Time = Time in Pit + Time for Acceleration
Total Time = 5.00 s + (-0.028 s)
Calculating the total time:
Total Time ≈ 4.97 s
Finally, we can find the distance the Mercedes Benz has traveled during this time by multiplying its constant speed (71.0 m/s) by the total time:
Distance = Speed * Time
Distance = 71.0 m/s * 4.97 s
Calculating the distance:
Distance ≈ 353.87 m
Therefore, the Thunderbird has fallen behind the Mercedes Benz by approximately 353.87 meters.