a train accelerates from rest to a speed v m/s. it continues at this constant speed for a certain time and then decelerates uniformly to rest. if the average speed for the whole journey is 5v/6, show that the 4/5 of the whole distance is covered at a constant speed.
i cant work this out. i am trying to make use of the formulas average speed = total distance/total time and s=t(u+v)/2 . i am working these out with the letters but it aint working out
uniform acceleration
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Re: uniform acceleration
First note that it doesn't matter what the acceleration/deceleration is, as long as the velocity graph is a trapezoid with a fixed ratio between the top and bottom sides. So, to me, it's easiest to consider an infinite deceleration, so the velocity graph becomes a triangle (the acceleration part) and a rectangle (the cruising part). It doesn't actually matter, since the area sum of two triangles with the same height is equal to the area of a triangle with the same height and the sum of their bases (and therefore the total distance and therefore the average velocity), so the following equation holds regardless.
So you spend a factor c time accelerating* and 1c at cruising speed (or vice versa), and their sum must equal the average velocity of the whole journey. From here on the next steps should be trivial.
*oh, in case my whole post confuses rather than clarifies: this does result in an average velocity. So you can remove the vs from the equation (if it's nonzero) and solve for c.
So you spend a factor c time accelerating* and 1c at cruising speed (or vice versa), and their sum must equal the average velocity of the whole journey. From here on the next steps should be trivial.
*oh, in case my whole post confuses rather than clarifies: this does result in an average velocity. So you can remove the vs from the equation (if it's nonzero) and solve for c.
Re: uniform acceleration
If the train accelerates uniformly from a speed of 0 to a speed of v, what is its average speed over this interval? If the train decelerates uniformly from a speed of v to a speed of 0, what is its average speed over this interval?
(∫p^{2})(∫q^{2}) ≥ (∫pq)^{2}
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Thanks, skeptical scientist, for knowing symbols and giving them to me.

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 Joined: Tue Oct 18, 2016 10:24 am UTC
Re: uniform acceleration
You have to tell the time periods? Otherwise, it will be one quantity will be expressed as other quantity.
This situation happens, when you provide less number of equations than number of variables that are directly involved (through any combination of equations).
This situation happens, when you provide less number of equations than number of variables that are directly involved (through any combination of equations).
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