Motion with constatnt acceleration problem

[Stochastic13]

Homework Statement

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

Homework Equations

x = xo + vot + 1/2 at^2 where xo and vo are initial distance and velocity respectively

The Attempt at a Solution

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?

 

[Stochastic13]

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.

 

[AndyUrquijo]

Hi.

You get the right answer following your problem statement. So wheter the rocket motor runs out of fuel or whatever the case. Saying it runs the last half of the track at constant speed is probably right.

You have to put the words into equations like this:

(1) "Starting from rest, the rocket motor was accelerated constantly for half the distance of the track..."

$$\frac{R_T}{2} = x_{or} + v_{or} t_{r1} + \frac{a_r t_{r1}^2}{2} = \frac{a_r t_{r1}^2}{2}$$

(2) " ...(The rocket motor) ran the other half at a constant speed."

$$\frac{R_T}{2} = v_{fr} t_{r2} = (a_r t_{r1}) t_{r2}$$

(3) "Next, a jet motor was started from rest and finished the whole track with constant acceleration for the entire distance."

$$R_T = x_{o j} + v_{oj} t_j + \frac{a_j t_j^2}{2} = \frac{a_j t_j^2}{2}$$

(4) "Both motors covered the same distance in the same time."

$$t_j = t_r = t_{r1} + t_{r2}$$

$$R_T$$ is the length of the track. $$t_{r1}$$ and $$t_{r2}$$ are the times the rocket motor spent on the first and second halves of the track respectively. $$x_or$$ , $$x_oj$$ , $$v_or$$ and $$v_oj$$ are the initial positions and velocities, all equal to zero.

Mess around with this equations and you will find the answer.

 

[Stochastic13]

Thanks, I really appreciate it, it worked out beautifully.

P.S. How did you write the equations like that? I only used my keyboard and that makes them hard to understand, next time I'd rather write the equations in clearly like you did to make it easier for others to see what I am doing.

 

[Stochastic13]

Nevermind I just figured it out, it didn't work for me earlier because I had the script blocked, just had to unblock it. Thanks for the help though, I just needed that last t1 + t2 equation that you showed and everything fell into place :)

 

[AndyUrquijo]

Glad to help.

This forum is $$\LaTeX$$-compatible. Check the last button with a sigma letter for some help. Actually, I recommend learning Latex altogether.

 

[Stochastic13]

Great, definitely looks like something useful. Thanks