Question

A rocket sled accelerates from rest on a level track with negligible air and rolling resistances. The initial mass of the sled is M0 = 600 kg. The rocket initially contains 150 kg of fuel. The rocket motor burns fuel at constant rate ˙m = 15 kg/s. Exhaust gases leave the rocket nozzle uniformly and axially at Ve = 2900 m/s relative to the nozzle, and the pressure is atmospheric. Find the maximum speed reached by the rocket sled. Calculate the maximum acceleration of the sled during the run.

Answer 1

a)
`v \approx 834.278\,(m)/(s)`, b)
`a = 96.667\,(m)/(s^(2))`

Step-by-step explanation:

a) The maximum speed of the rocket is given by the Tsiolkovski's Equation:

`v =v_(o) - v_(ext)\cdot \ln \left((m)/(m_(o)) \right)`

`v = 0\,(m)/(s) - (2900\,(m)/(s) )\cdot \ln \left((450\,kg)/(600\,kg) \right)`

`v \approx 834.278\,(m)/(s)`

b) The acceleration is obtained by deriving the Tsiolkolski's Equation:

`a = -v_(ext)\cdot \left((1)/(m)\left) \cdot \dot m`

The maximum acceleration occured when fuel is entirely consumed. Then:

`a = - \left(2900\,(m)/(s) \right)\cdot \left((1)/(450\,kg) \right)\cdot \left(-15\,(kg)/(s) \right)`

`a = 96.667\,(m)/(s^(2))`