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.

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To find the time it takes for the runner to cross the finish line, we can use the kinematic equation:

d = v0t + 0.5at^2

Where:
d = distance traveled (100 m)
v0 = initial velocity (0 m/s, since the runner starts from rest)
a = acceleration (0.55 m/s^2)
t = time it takes to cross the finish line (unknown)

Rearranging the equation to solve for time:

t = (sqrt(2ad + v0^2) - v0) / a
t = (sqrt(2 * 0.55 * 100 + 0^2) - 0) / 0.55
t = (sqrt(110) - 0) / 0.55
t ≈ 11.47 s

Therefore, it takes the runner approximately 11.47 seconds to cross the finish line.

To find the time it takes for the runner to return to rest after crossing the finish line, we can consider the final velocity.

Using the equation:

v = v0 + at

Where:
v = final velocity (0 m/s, since the runner comes to rest)
v0 = initial velocity (unknown)
a = acceleration (-0.55 m/s^2, since the runner is decelerating)
t = time it takes to return to rest (6.5 s)

Rearranging the equation to solve for initial velocity:

v0 = v - at
v0 = 0 - (-0.55 * 6.5)
v0 = 3.57 m/s

Therefore, the initial velocity of the runner is approximately 3.57 m/s before he begins slowing down.

To summarize:
- It takes the runner approximately 11.47 seconds to cross the finish line.
- The initial velocity of the runner before slowing down is approximately 3.57 m/s.
- It takes the runner 6.5 seconds to return to rest after crossing the finish line.

To solve this problem, we can break it down into two parts: the runner's motion while crossing the finish line, and the runner's motion after crossing the finish line until coming to a stop.

Let's first consider the runner's motion while crossing the finish line.

1. Calculate the time it takes for the runner to cross the finish line (tr):
We can use the equation of motion, assuming constant acceleration:
d = vi * t + (1/2) * a * t^2
Since the runner starts from rest (vi = 0), the equation simplifies to:
d = (1/2) * a * t^2
Rearranging the equation, we get:
t = √(2 * d / a)
Substituting the given values: d = 100 m and a = 0.55 m/s^2, we can solve for tr.

Next, let's consider the runner's motion after crossing the finish line until coming to a stop.

2. Calculate the runner's final velocity (vf) after crossing the finish line:
We can use the equation of motion:
vf = vi + a * t
Since the runner crosses the finish line with constant acceleration, his final velocity at that point is:
vf = 0 + (a * tr)
Substituting the previously calculated value of tr, we can find vf.

3. Calculate the time it takes for the runner to come to a stop after crossing the finish line (ts):
Using the equation of motion:
vf = vi + a * t
Rearrange the equation to solve for t:
t = (vf - vi) / a
Since the runner comes to a stop (vf = 0), the equation becomes:
0 = vi + a * ts
Rearrange the equation to solve for ts:
ts = -vi / a
Substituting the previously calculated value of vf, we can find ts.

By solving these equations, we can find the values of tr and ts, which will give us the time it takes for the runner to cross the finish line and the time it takes for the runner to come to a stop after crossing the finish line, respectively.