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.

during the run:

vf^2=vi^2 + 2ad solve for vf

but vf= ai*t solve for time t.
so sqrt (2ad)=ai*t
square both side
2ad=a^2 t^2
t= sqrt (2d/a)

Well, to find the expression for the time it takes for the runner to cross the finish line, tr, let's start by looking at the motion of the runner.

From the given information, we know that the runner starts with an initial acceleration ai of 0.55 m/s^2 and crosses the finish line at a distance of 100 m.

Using the second equation of motion, we can relate the distance, acceleration, and time as follows:

d = ut + (1/2)at^2

Since the runner starts from rest (u = 0), the equation simplifies to:

d = (1/2)at^2

By substituting the given values, we get:

100 = (1/2)(0.55)t^2

Now, let's solve for t:

200 = 0.55t^2

Dividing both sides by 0.55:

t^2 = 200/0.55

t^2 ≈ 363.64

Taking the square root of both sides:

t ≈ √363.64

t ≈ 19.08 seconds

So, the expression for the time it takes for the runner to cross the finish line, tr, in terms of d and ai is approximately 19.08 seconds.

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 kinematic equation:

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

Since the runner starts from rest (vi = 0), the equation simplifies to:

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

Now, we can rearrange the equation to solve for time, t:

t^2 = (2*d) / ai

Taking the square root of both sides:

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

Therefore, the expression for the time it takes for the runner to cross the finish line, tr, is:

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

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

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

Since the runner starts from rest (vi = 0) at the starting gate, the equation becomes:

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

Rearranging the equation, we can solve for t:

2d = ai * t^2

Dividing both sides by ai gives us:

(2d)/ai = t^2

Taking the square root of both sides:

t = sqrt((2d)/ai)

So, the expression for the time it takes for the runner to cross the finish line, tr, from the start in terms of distance (d) and acceleration (ai) is:

tr = sqrt((2d)/ai)