Im having this problem with a rocket equation. Ill state the problem then show what i've done(adsbygoogle = window.adsbygoogle || []).push({});

Let [tex]M[/tex] = mass of rocket and fuel

[tex]M_f[/tex] = mass of fuel

[tex]M_0[/tex] = rockets total initial mass (including fuel)

(this is given as [tex]10^5 kg[/tex])

[tex]V_R[/tex] = rockets velocity

A [tex]10^5 kg[/tex] rocket has a total weight-to-payload ratio of 10 to 1. i.e [tex]M_f = 9M_0[/tex]

The rocket starts from rest, pointing vertically. Its exhaust gases are expelled at a rate [tex]k (=450 kg/s)[/tex] with a constant velocity [tex]v_0(= 2,600 m/s)[/tex]. Assume the rocket experiences a constant gravitational force with [tex]g = 9.8 m/s^2[/tex]

Work out the answers to the following questions using the above symbols together with [tex]k, v_0[/tex] and [tex]g[/tex]. Put in numerical values only at the end.

(1) What is the burn-out time?

(2) What is the rockets final velocity, assuming it rises vertically during the boost phase?

(3) How high is the rocket at the end of its boost phase?

Ok, i've done (1) and am trying to do (2). Im trying to set up the differential equation of the rockets motion.

I said that at [tex]t = 0[/tex]

Total momentum = 0

then at [tex]t = dt[/tex]

[tex]dp_{tot} = (M - dM_f)(dV_R) - (v_0dM_f)[/tex]

neglecting 2-nd order terms

[tex]\displaystyle{\frac{dp_{tot}}{dt} = M\frac{dV_R}{dt} - v_0\frac{dM_f}{dt}}[/tex]

Then as the only external force on the rocket is gravity

[tex]\displaystyle{-(M - dM_f)g = M\frac{dv_R}{dt} - v_0\frac{dM_f}{dt}}[/tex]

I also know that [tex]\displaystyle{\frac{dM_f}{dt} = -k[/tex]

so

[tex]\displaystyle{-(M - dM_f)g = M\frac{dv_R}{dt} + v_0k}[/tex]

Is this at all right so far?

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# Homework Help: Rocket Motion

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