On a horizontal test track two motors (jet and rocket) are tested.  Starting from rest, the rocket motor was accelerated constantly for half the distance of the track and ran the other half at a constant speed.  Next, a jet motor was started from rest and finished the whole track with constant acceleration for the entire distance.  Both motors covered the same distance in the same time.  Show that the ratio of the acceleration of the jet motor to rocket motor is given by:  aj/ar = 8/9      I wrote two equations for the distance of each motor.  For rocket: x = 1/8 ar*t^2 + v*t/2 and since v*t/2 = ar*t^2/4  we get  x = 1/8 ar*t^2 + v*t^2/4 and for the jet motor: x = 1/2 aj*t^2 and set them equal to each-other and find that the ratio is  aj/ar = 3/4  instead of 8/9.  What am I doing wrong?

Edited March 30, 201114 yr by CaptainBlood

I just got an answer from popovoleg that makes a lot of sense, that acceleration remains constant. In my original question I said that the other half the distance is traveled at a constant speed; I should have said that the rocket runs out of fuel at half the distance for the rocket motor. But I then get (1/2) aj * t^2 = (1/2) ar (t/2)^2 and the final answer comes out aj / ar = 1/4 which is not 8/9 so I'm still making a mistake somewhere.

 

Ok, I did it, I just needed to use t1 for the first time which was at1 and t2 for the second so the equation I was missing is t1 + t2 and everything works out great.