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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 - Differential Equation

### Rocket Sled - Integrate the Rocket Equation

### Rocket Sled - Solve for Initial Velocity

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