Relationship Between Velocities of Two Runners

[Drakkith]

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_j, so that each has the same displacement during the time interval. Note: t _f is NOT twice t _max , but represents an arbitrary time. What is relation between V_j and t_max?

Homework Equations

Position equation for constant acceleration: X=X_i+V_i+1/2AT²
Velocity Equation for constant acceleration: V=V_i+AT
Position Equation for constant velocity: X=X_i+VT

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_jt_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: X₁ = 1/2At_max²
From t_max to t_f: X = X₁ + V_maxt_f + 1/2At_f²Since X = V_jt_f, we can rewrite the above equation as: V_jt_f = X₁ + V_maxt_f + At_f²
Replacing X₁ with its equation: V_jt_f = 1/2At_max²+ V_maxt_f + At_f²

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

 

[jbriggs444]

Distance traveled is the integral of velocity over time -- the area under the velocity versus time graph. Given that, the problem turns into a simple geometric exercise. One needn't bother with any equations at all.

 

[Drakkith]

Distance traveled is the integral of velocity over time -- the area under the velocity versus time graph. Given that, the problem turns into a simple geometric exercise. One needn't bother with any equations at all.

Thanks Jbriggs. Apparently the question was asking about Vmax, not Tmax, as V_j = 1/2 V_max is the correct answer. (Which wouldn't have helped me prior to your post anyways, I still had no idea to look for the area under the graph)

Edit: Just so I don't look like an idiot, the question I posted was a literal copy and paste. I didn't just misread V_max as T_max.