8.01SC | Fall 2016 | Undergraduate

# Classical Mechanics

Week 6: Continuous Mass Transfer

## PS.6.1 Rocket Sled Problem

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A rocket sled can eject gas backwards or forwards at a speed $$\displaystyle u$$ relative to the sled. The initial mass of the fuel in the sled is equal to dry mass of the sled, $$\displaystyle m_0$$. At $$\displaystyle t=0$$ the rocket sled has speed $$\displaystyle v_0$$ and starts to eject fuel in the forward direction in order to slow down. You may ignore air resistance. You may treat $$\displaystyle u$$ as a given constant in the following questions.

(Part a) Let $$\displaystyle v_ r(t)$$ be the x-component of the rocket sled velocity, and $$\displaystyle m_ r(t)$$ the mass of the rocket, dry mass and the fuel inside the rocket, at a given time $$\displaystyle t$$. Derive differential equation for $$\displaystyle v_ r$$ in terms of $$\displaystyle m_ r$$.

(Part b) Integrate the equation you derived for $$\displaystyle v_ r$$ to find the velocity of the rocket sled as a function of mass, $$\displaystyle v_ r(m_ r)$$, as the rocket sled slows down. Write your answer using some or all of the following: $$\displaystyle u$$, $$\displaystyle v_0$$,$$\displaystyle m_0$$, and $$\displaystyle m_ r$$.

(Part c) What was the initial speed $$\displaystyle v_0$$ of the rocket sled if the sled came to rest just as all the fuel was burned? Write your answer using some or all of the following: $$\displaystyle u$$, $$\displaystyle m_0$$, and $$\displaystyle m_ r$$.

### Rocket Sled - Solve for Initial Velocity

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