Rocket Motion with Fnet=0 (on the rocket/exhaust system)

Example

The V2 Rocket was the world's first long-range ballistic missile.

total mass = 12500 kg

mass of propellant = 8720 kg

mass of rocket body = 3780 kg

exhaust speed = 2000 m/s

thrust = 265,000 N

Model the motion of the rocket numerically. Graph $v_x(m)$ and $v_x(t)$.
If this rocket is in space and starts from rest, what would be its maximum speed?
How long will it take to burn all of its fuel?
What was its displacement between where it started and its location when it reaches its maximum speed?



In [1]:

    
%matplotlib inline



In [2]:

    
from __future__ import division, print_function
from numpy import *
from ivisual import *
import matplotlib.pyplot as plt



In [3]:

    
mfuel=8720
mbody=3780
m0=mfuel+mbody
m=mfuel+mbody
mf=mbody
vexhaust=2000
thrust=2.65e5
mdot=-thrust/vexhaust

r=vector(0,0,0)
v=vector(0,0,0)

t=0
dt=0.005

tlist=[]
xlist=[]
vxlist=[]
mlist=[]
vtheorlist=[]

while m>mf:
    Fthrust=vector(thrust,0,0)
    Fnet=Fthrust
    m=m+mdot*dt
    
    v=v+Fnet/m*dt
    r=r+v*dt
    
    vtheor=vexhaust*log(m0/m)
    
    t=t+dt
    
    tlist.append(t)
    xlist.append(r.x)
    vxlist.append(v.x)
    mlist.append(m)
    vtheorlist.append(vtheor)

    
print("burnout at t=",t," s") 
print("velocity at burnout = ",v.x," m/s")
print("mass at burnout = ",m," kg")
print("displacement at burnout=",r.x," m")









    



burnout at t= 65.815  s
velocity at burnout =  2392.38951431  m/s
mass at burnout =  3779.5125  kg
displacement at burnout= 63399.9538405  m



In [4]:

    
plt.title('vx vs m')
plt.xlabel('m (kg)')
plt.ylabel('vx (m/s)')
plt.plot(mlist,vxlist,'y.',mlist,vtheorlist,'r-')
plt.show()



In [5]:

    
plt.title('vx vs t')
plt.xlabel('t (s)')
plt.ylabel('vx (m/s)')
plt.plot(tlist,vxlist,'b.')
plt.show()



In [6]:

    
plt.title('x vs t')
plt.xlabel('t (s)')
plt.ylabel('x (m)')
plt.plot(tlist,xlist,'b.')
plt.show()



In [6]: