Rockets and fuel expulsion

Really struggling with this one. In class we used velocity not speed for starters and had $m$ and $m+\delta t$. I cant figure out how to relate it to this question at all. If someone could point me in the right direction that would be great.

A car is propelled by a rocket engine along a smooth straight horizontalroad. Initially, the car is at rest and has mass M0. The spent fuel is expelledwith a constant speed c relative to the car. Show that the kinetic energy ofthe car when all the fuel has been used up is

What proportion of the initial mass M0 should the initial mass of fuel be inorder to maximise the kinetic energy of the car when all the full has been usedup?

[Hint for last part: note that the kinetic energy will be zero if either M1 = 0 orM1 = M0, so to find the value of M1 for which the kinetic energy is maximisedas M1 varies between 0 and M0, you need to differentiate the kinetic energywith respect to M1]

Re: Rockets and fuel expulsion

You can use impulse and momentum principles:

Rearrange and integrate:

where = initial total mass of ship plus fuel, and = final mass of ship after all fuel is spent.

Now that you have a value for , you can calculate its final kinetic energy.

To answer the second part it would be best to introduce a variable for the ratio of the ship's mass to the ship + fuel mass. Let , and the KE equation becomes

Now find the value of k that maximizes KE by setting .

Re: Rockets and fuel expulsion

Quote:

Originally Posted by

**ebaines** You can use impulse and momentum principles:

Rearrange and integrate:

where [tex] M_0 {/tex] = initial total mass of ship plus fuel, and

= final mass of ship after all fuel is spent.

Now that you have a value for

, you can calculate its final kinetic energy.

To answer the second part it would be best to introduce a variable for the ratio of the ship's mass to the ship + fuel mass. Let

, and the KE equation becomes

Now find the value of k that maximizes KE by setting

.

Hi, thank you very much for the reply. I have completed the first part and understand the secong (although I didnt think of a substitution) but I'm struggling to differentiate the equation with respect to m1

Re: Rockets and fuel expulsion

If you treat as a variable and as a constant, then you can proceed as follows:

Now recall that the derivarive of , and also that the dierivative of . So this becomes:

Set this equal to zero and solve for . What do you get?

Re: Rockets and fuel expulsion

Iget $M_1$ to be or does that mean the energy is max when its at ?

Re: Rockets and fuel expulsion

Yes, that's correct. And so the proportion of M_0 that should be fuel is .