Mechanics Problem, Cannot understand

[mathwurkz]

Here is the problem.

A car starts moving rectilinearly, first with acceleration $$\omega = 5.0$$ m/s^2 (the initial velocity is equal to zero), then uniformly, and finally, decelerating at the same rate $$\omega$$, comes to a stop. The total time of motion equals $$\tau = 25$$ s. The average velocity during that time is equal to $$v = 72$$ km per hour. How long does the car move uniformly?

If I did not have to deal with the average velocity, then I can figure out that the car will reach a maximum speed of 50 m/s for an instant before it has to decelerate in order to have a 25 s trip. But I think this is where I have a weakness when it comes to average velocities. How do you put it together with acceleration? The answer at the back of my book gives a result of 15 s. I'd appreciate any help getting me pointed in the right direction.

 

[Fermat]

create equations of motion for all three stages of the journey.

For the distance traveled and for the velocities at the end-points of each stage.

Remember that the distance traveled and time taken are the same for the 1st and 3rd stages.

Edit: I got 15s too.

 

[mcah5]

Pretty much what Fermat said:

Find the distance traveled during the first and second states. Note the distance and time displaced during the 3rd stage is the same as the first. So 2*x1+x2 = 500 m and 2t1 + t2 = 25 s. Find the equations for x1 and x2 in terms of t1 and t2, then solve the system of equations.

 

[mathwurkz]

Thank you I finally solved it. However, what still puzzles me is what the answer at the back gives. I don't understand their interpretation. Then again I don't know if it is even correct since I think there would be a neagtive square root.
What the book gives me:

$$\Delta t = \tau \sqrt{\frac{1-4v}{\omega \tau}} = 15 s$$