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. 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.

Sqrt(2d/ai)

To find an expression for the time it takes for the runner to cross the finish line, tr, in terms of d (distance) and ai (constant acceleration), we can use the kinematic equation:

d = vi * t + (1/2) * a * t^2

Where:
- d is the distance covered by the runner (100 m in this case)
- vi is the initial velocity of the runner when he starts running (0 m/s, as he starts from rest)
- a is the acceleration of the runner (0.55 m/s^2)
- t is the time it takes for the runner to cross the finish line (tr, the value we want to find)

Since the runner starts from rest, vi = 0, and the equation becomes:

d = (1/2) * a * t^2

Rearranging the formula to solve for t, we get:

t^2 = (2 * d) / a

Taking the square root of both sides, we can solve for t:

t = sqrt((2 * d) / a)

Substituting the given values, we get:

t = sqrt((2 * 100 m) / 0.55 m/s^2)

Simplifying further, we have:

t = sqrt(363.636 s^2/m)

Therefore, the expression for the time it takes for the runner to cross the finish line, tr, in terms of d and ai is:

tr = sqrt((2 * d) / ai)