A runner is at the starting gate and hears the starting gun. He begins running with a constant acceleration ai = 0.55 m/s2. He crosses the finish line at d = 100 m and then begins slowing down. It takes him tr to cross the finish line. It takes him ts = 6.5 s to return to rest after crossing the finish line. For this problem, use a coordinate system with the runner is moving in the positive direction.
Randomized Variables
ai = 0.55 m/s2
ts = 6.5 s
show answer No Attempt 33% Part (a) Find an expression for the time it takes for the runner to cross the finish line, tr, from the start in terms of d and ai.
To find an expression for the time it takes for the runner to cross the finish line, tr, from the start in terms of d and ai, we can use the equations of motion.
Let's use the following equations:
1. vf = vi + at (final velocity equals initial velocity plus acceleration multiplied by time)
2. d = vi*t + (1/2)*a*t^2 (distance equals initial velocity multiplied by time plus half the acceleration multiplied by the square of time)
Initially, the runner is at rest and starts running with a constant acceleration.
First, we need to find the initial velocity of the runner, vi.
Using equation 1, we know that at the finish line, vf (final velocity) is 0 m/s (the runner comes to rest). The initial velocity, vi, is unknown, and the acceleration is given by ai = 0.55 m/s^2.
0 = vi + ai*ts
Rearranging the equation, we get:
vi = -ai*ts
Now, let's find the time it takes for the runner to cross the finish line, tr.
Using equation 2, we know that the distance, d, is 100 m, the initial velocity, vi, is -ai*ts, and the acceleration is given by ai = 0.55 m/s^2.
d = vi*tr + (1/2)*ai*tr^2
Substituting the values, we have:
100 = (-ai*ts)*(tr) + (1/2)*ai*(tr)^2
Solving this equation will give the expression for the time it takes for the runner to cross the finish line, tr, from the start in terms of d and ai.