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DYNAMICS - CASE STUDY SOLUTION


Force and Acceleration Diagram
of Rocket

 

Use Newton's Second Law to sum all the forces acting on the rocket and equate it to the acceleration times mass,

     ΣF = ma

     Ff - mg = m (dv/dt)

The thrust, Ff, is due to the mass flow of the burning propellant. Using the mass flow equation gives,

     Ff = -(dmf /dt) vf = (dmf /dt) vfj = cvfj

Notice, all motion and forces are only in the y direction. Combining equations for the y-direction,

     cvf - mg = m (dv/dt)

Rearrange to get

     dv = (cvf /m - g)dt

At any time t, the total mass of the rocket and the remaining fuel can be expressed as

     m = mo - (dmf /dt) t = mo - ct

By substituting, the change in velocity, dv, is

     

     


Plot of Rocket Velocity Equation

 

Integration will give an expression for the velocity as

     

The total time required to burn all the fuel is

     mf = (dmf /dt)ttot = c ttot

     ttot = mf /c

Now that the time needed to burn all the fuel is known, it can be substituted back into the velocity equation to give,

     

where vf = fuel velocity (constant)
         mo = initial mass of rocket and fuel
         mf = initial mass of fuel
         c = fuel burn rate, dmf/dt

     
   
 
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