Mechanics Problem, Cannot understand

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In summary, the car travels rectilinearly with an initial acceleration of 5.0 m/s^2 and reaches a maximum speed of 50 m/s before decelerating at the same rate and coming to a stop in 25 s. The average velocity during this time is 72 km/h. To find the time of uniform motion, we can set up equations for the distance and time traveled during the first and second stages and solve for the system of equations. The result is 15 s.
  • #1
mathwurkz
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Here is the problem.

A car starts moving rectilinearly, first with acceleration [tex] \omega = 5.0 [/tex] m/s^2 (the initial velocity is equal to zero), then uniformly, and finally, decelerating at the same rate [tex] \omega [/tex], comes to a stop. The total time of motion equals [tex] \tau = 25 [/tex] s. The average velocity during that time is equal to [tex] v = 72 [/tex] 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.
 
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  • #2
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.
 
  • #3
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.
 
  • #4
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:

[tex] \Delta t = \tau \sqrt{\frac{1-4v}{\omega \tau}} = 15 s[/tex]
 

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Mechanics problem refers to a problem or question related to the study of motion and forces, and how they affect the behavior of objects.

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