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## Homework Statement

Reginald is out for a morning jog, and during the course of his run on a straight track, he has a velocity that depends upon time as shown in the figure below. That is, he begins at rest, and ends at rest, peaking at a maximum velocity

*V*

_{max}at an arbitrary time

*t*

_{max}. A second runner, Josie, runs throughout the time interval

*t*= 0 to

*t*=

*t*

_{f}at a constant speed

*V*, so that each has the same displacement during the time interval. Note:

_{j}*t*

_{f}is NOT twice

*t*

_{max}, but represents an arbitrary time. What is relation between

*V*and

_{j}*t*

_{max}?

## Homework Equations

Position equation for constant acceleration: X=X

_{i}+V

_{i}+1/2AT

^{2}

Velocity Equation for constant acceleration: V=V

_{i}+AT

Position Equation for constant velocity: X=X

_{i}+VT

3. The Attempt at a Solution

3. The Attempt at a Solution

Not sure what to do really, so I've just been trying things.

The only commonality between the two runners is the final position X and the time t

_{f}

For Josie, X = V

_{j}t

_{f}

For Reginald:

Velocity:

From t

_{i}to t

_{max}, V

_{max}= At

_{max}

From t

_{max}to t

_{f}, since V

_{f}= 0, the equation is: 0 = V

_{max}+ At

_{f}

Position:

From t

_{i}to t

_{max}, initial displacement and velocity are zero: X

_{1}= 1/2At

_{max}

^{2}

From t

_{max}to t

_{f}: X = X

_{1}+ V

_{max}t

_{f}+ 1/2At

_{f}

^{2}Since X = V

_{j}t

_{f}, we can rewrite the above equation as: V

_{j}t

_{f}= X

_{1}+ V

_{max}t

_{f}+ At

_{f}

^{2}

Replacing X

_{1}with its equation: V

_{j}t

_{f}= 1/2At

_{max}

^{2}+ V

_{max}t

_{f}+ At

_{f}

^{2}

That's about as far as I've gotten and I don't know if I'm even on the right track.