8.01SC | Fall 2016 | Undergraduate

# Classical Mechanics

## Week 6: Continuous Mass Transfer

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### Week 6 Problem Set

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Problem Set 6 contains the following problems:

1. Rowing Across the River
2. Astronauts Playing Catch
3. Multi-stage Rocket in Empty Space
4. Falling Drop
5. Moving Vehicle and Falling rain

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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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Snow of density $$\displaystyle \rho$$ covers a road to a uniform depth of $$\displaystyle D$$ meters. A snowplowing truck of mass $$\displaystyle M$$ starts clearing the road at $$\displaystyle t = 0$$ at an initial velocity $$\displaystyle v_0$$. The contact between the tires and the road applies a constant force $$\displaystyle F_0$$ in the forward direction. The truck’s subsequent velocity depends on time as it clears a path of width $$\displaystyle W$$ through the snow. The snow, after coming momentarily to rest relative to the truck, is ejected sideways, perpendicular to the truck.

(Part a) Find a differential equation relating the change in the velocity of the truck $$\displaystyle dv/dt$$ to its velocity $$\displaystyle v(t)$$. Express you answer in terms of some or all of the following: $$\displaystyle \rho$$, $$\displaystyle D$$, $$\displaystyle W$$, $$\displaystyle M$$, $$\displaystyle v$$ and $$\displaystyle F_0$$.

(Part b) Calculate $$\displaystyle v_{\text {term}}$$, the terminal speed reached by the truck. Express you answer in terms of some or all of the following: $$\displaystyle \rho$$, $$\displaystyle D$$, $$\displaystyle W$$, $$\displaystyle M$$ and $$\displaystyle F_0$$.

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Fall 2016
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