two runners are competing in a 400-meter dash. The first runs at 6m/s and maintains this speed. The second runner starts at 10m/s and decelerates at 1.8m/s^2. Who will finish first? What will be their time difference?
d = Vi t + .5 a t^2
runner 1
400 = 6 t and a = 0
t = 400/6
runner 2
400 = 10t - 0.9 t^2
.9 t^2 - 10 t + 400 = 0 solve quadratic
t = complex number, never makes it :)
The velocity turns negative before the runner reaches the finish
Let's see when v = 0
v = 10 -1.8 t
v = 0 when t = 10/1.8 = 5.6 seconds
then
d = 10 (5.6) - 0.9 (5.6)^2 = 56 - 28.2 = 27.8
so this second runner starts backing up after 28 meters :)
To determine who will finish first and the time difference between the two runners, we need to calculate the time taken by each runner to complete the 400-meter dash.
Let's start by calculating the time taken by the first runner:
Distance = Speed × Time
400 meters = 6 m/s × Time
Solving for Time:
Time = Distance / Speed
Time = 400 meters / 6 m/s
Time = 66.67 seconds (rounded to two decimal places)
Now, let's calculate the time taken by the second runner:
Since the second runner is decelerating at a rate of 1.8 m/s^2, we need to calculate the constant speed he will reach before the finish line.
Using the equation:
Final Speed^2 = Initial Speed^2 + 2 × Acceleration × Distance
Final Speed^2 = 10 m/s^2 + 2 × (-1.8 m/s^2) × 400 meters
Final Speed^2 = 10 m/s^2 + (-3.6 m/s^2) × 400 meters
Final Speed^2 = 10 m/s^2 - 1440 m/s^2
Final Speed^2 = -1430 m/s^2
Since speed cannot be negative, the second runner will not reach the finish line. Therefore, the first runner will finish the race first.
The time difference between the two runners will be equal to the time it takes for the first runner to complete the race, which is 66.67 seconds.