(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

So if a rocket is launched vertically from the surface of the earth, the rocket has a mass of 1000 kg and has a fuel load of 12000 kg. The fuel burns at such a rate that it will be exhausted after 180 secs. The exhaust velocity of the burned fuel relative to the rocket is 2000 m/sec. Assume an air resistance proportional to the velocity with proportionality constant b = 100 n-sec/m and that the gravity is constant. Find the velocity and height of the rocket when the fuel runs out.

2. Relevant equations

Force = mass * acceleration = mass * gravity - air resistance * velocity\

Mass = 13000 (weight of rocket and fuel) - 66.66t (weight of fuel lost per second or 12000/180)

3. The attempt at a solution

So, naturally, I want to find the velocity with the above equation. So with inserting the known values into the equation it becomes,

(13,000 - 66.66t)a = (13,000 - 66.66t) * 9.8 (gravitational constant - 100 (air resistance) * v

dividing mass to the other side I get.

a = 9.8 - (100 * v) / (13,000 - 66.66t)

Since I will have to integrate a in get it in terms of v, I divided the right half to the left half

dv/dt (acceleration) / 9.8 - (100 * v) / (13,000 - 66.66t) = 1 dt

Integrate both sides (I had to use a program to solve this)

(-6666t - 130)ln(-653.268t - 100v + 127,400) = t

now we have to get v by itself so,

ln (-653.268t - 100v + 127,400) = t / (-6666t - 130)

Get rid of the ln,

-653.269t - 100v + 127,400 = e ^ (t / (-6666t - 130)

more simple rearranging gets

-100v = e ^ (t / (-6666t - 130) + 653.269t - 127,400

v = -(e ^ (t / (-6666t - 130) + 653.269t - 127,400) / 100

now we have to subtract the exhaust velocity from all this equaling

v = (-(e ^ (t / (-6666t - 130) + 653.269t - 127,400) / 100) - 2000

This should give the velocity of the rocket when the fuel runs out. I theorize that to find the height you simply have to take the integral of this function, and since I don't want to re-write that mess (at least not without knowing if I am actually correct in that assumption), I would greatly appreciate if someone could analyze this work and see if I both found the correct velocity and am correct in how to find the height. Thanks in advance for your help.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: The velocity and height of a rocket launched vertically from the earth.

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**