A rocket is moving vertically upward against gravity. Its mass at time t is `m=m_0-mut` and it expels burnt fuel at a speed u vertically downward relative to the rocket. Derive the equation of motion of the rocket but do not solve it. Here, `mu` is constant.

Explanation

To derive the equation of motion for the rocket moving vertically upward against gravity, we will follow these steps:

### Step 1: Identify the forces acting on the rocket
The forces acting on the rocket are:
1. The thrust force \( F_t \) due to the expulsion of burnt fuel.
2. The weight of the rocket \( W = mg \), where \( g \) is the acceleration due to gravity.

### Step 2: Write the expression for the mass of the rocket
The mass of the rocket at time \( t \) is given by:
\[ m(t) = m_0 - \nu t \]
where \( m_0 \) is the initial mass of the rocket and \( \nu \) is the rate at which the mass decreases (burnt fuel).

### Step 3: Apply Newton's second law
According to Newton's second law, the net force \( F \) acting on the rocket is equal to the mass of the rocket multiplied by its acceleration \( a \):
\[ F = m(t) a \]
The net force can also be expressed as the thrust force minus the weight:
\[ F = F_t - mg \]

### Step 4: Substitute the expressions into Newton's second law
Substituting the expressions for mass and forces into Newton's second law gives:
\[ F_t - mg = m(t) a \]
Substituting \( m(t) = m_0 - \nu t \) and \( mg = (m_0 - \nu t)g \):
\[ F_t - (m_0 - \nu t)g = (m_0 - \nu t) a \]

### Step 5: Express acceleration in terms of velocity
The acceleration \( a \) can be expressed as the time derivative of velocity \( v \):
\[ a = \frac{dv}{dt} \]
Thus, we can rewrite the equation as:
\[ F_t - (m_0 - \nu t)g = (m_0 - \nu t) \frac{dv}{dt} \]

### Step 6: Rearranging the equation
Rearranging the equation gives:
\[ (m_0 - \nu t) \frac{dv}{dt} = F_t - (m_0 - \nu t)g \]

### Step 7: Final form of the equation of motion
The final form of the equation of motion for the rocket can be expressed as:
\[ (m_0 - \nu t) \frac{dv}{dt} = F_t - (m_0 - \nu t)g \]

### Summary
The derived equation of motion for the rocket is:
\[ (m_0 - \nu t) \frac{dv}{dt} = F_t - (m_0 - \nu t)g \]