2 runners accelerate from rest to their top speeds. The 1st runner pat has an acceleration of 5.9 m/s^2 and reaches her top speed in 4 sec. The second runner steven accelerates at 4.4m/s^2 and takes 6.5 sec to reach his top speed the 2 runners run flat out for the remainder of the race they stay at their top speed if the race is the 200m who wins and with what time do they finish?
v1=a1•t1=5.9•4=23.6 m/s
v2= a2•t2 = 4.4•6.5 = 28.6 m/s
τ1 =s/v1 =200/23.6=8.47 s.
τ2 =s/v2 =200/28.6=6.99 s
the total time
for the 1st runner 4+8.47 = 12.47 s.
fir the 2nd runner 6.5+6.99 = 13.49 s
To determine who wins the race and their finishing times, we need to calculate the time it takes for each runner to reach their top speed and then compute the time it takes for them to cover the remaining distance at their top speed.
1. Calculate the time it takes for the 1st runner, Pat, to reach her top speed:
Pat's acceleration = 5.9 m/s^2
Time to reach top speed for Pat = 4 seconds
2. Calculate the time it takes for the 2nd runner, Steven, to reach his top speed:
Steven's acceleration = 4.4 m/s^2
Time to reach top speed for Steven = 6.5 seconds
3. Calculate the distance covered by each runner during their acceleration phase:
During the acceleration phase, the distance covered can be calculated using the formula:
Distance = (1/2) * acceleration * (time)^2
For Pat:
Distance covered by Pat during acceleration = (1/2) * 5.9 * (4^2) = 47.2 meters
For Steven:
Distance covered by Steven during acceleration = (1/2) * 4.4 * (6.5^2) = 89.9 meters
4. Calculate the remaining distance each runner needs to cover:
Remaining distance for both runners = Total race distance - Distance covered during acceleration phase
Total race distance = 200 meters
For Pat:
Remaining distance for Pat = 200 - 47.2 = 152.8 meters
For Steven:
Remaining distance for Steven = 200 - 89.9 = 110.1 meters
5. Calculate the time it takes for each runner to cover the remaining distance at their top speed:
Time = Distance / Speed
For Pat:
Time taken by Pat to cover remaining distance = 152.8 / Pat's top speed
For Steven:
Time taken by Steven to cover remaining distance = 110.1 / Steven's top speed
6. Determine the winner and their finishing time:
Compare the total time taken by each runner (acceleration time + remaining distance time).
If Pat has the shorter total time, she wins the race. Otherwise, Steven wins.
To find the finishing time for the winner, add the acceleration time to the time taken to cover the remaining distance.
7. Substitute the values of acceleration, time, and distance into the above formulas to calculate the exact times for both runners and determine the winner.
By following these steps, you can determine who wins the race and their finishing time.