Suppose a rocket with an intial mass `M_(0)` eject a mass `Deltam` in the form of gases in time `Deltat`, then the mass of the rocket after time t is :-

Options

A. `M_(0)-(Deltam)/(Deltat).t`
B. `M_(0)-(Deltam)/(Deltat)`
C. `M_(0)-(Deltam)/(Deltat)`
D. `M_(0)`

Answer

To solve the problem, we need to determine the mass of the rocket after a certain time `t` given that it has an initial mass `M_0` and ejects a mass `Δm` in a time interval `Δt`.

### Step-by-Step Solution:

1. **Identify Initial Mass**:
The initial mass of the rocket is given as \( M_0 \).

2. **Determine Mass Ejected**:
The mass ejected by the rocket in the time interval \( Δt \) is \( Δm \).

3. **Calculate Mass Ejection Rate**:
The mass ejection rate can be defined as:
\[ \text{Mass rate} = \frac{Δm}{Δt} \]
This gives us the amount of mass ejected per unit time.

4. **Find Total Mass Ejected After Time \( t \)**:
If we want to find out how much mass has been ejected after a time \( t \), we can multiply the mass ejection rate by the time \( t \):
\[ \text{Mass ejected after time } t = \left(\frac{Δm}{Δt}\right) \cdot t \]

5. **Calculate Remaining Mass of the Rocket**:
The remaining mass of the rocket after time \( t \) can be found by subtracting the mass ejected from the initial mass:
\[ M(t) = M_0 - \left(\frac{Δm}{Δt} \cdot t\right) \]

6. **Final Expression for Mass After Time \( t \)**:
Therefore, the mass of the rocket after time \( t \) is:
\[ M(t) = M_0 - \frac{Δm}{Δt} \cdot t \]

### Conclusion:
The mass of the rocket after time \( t \) is given by the formula:
\[ M(t) = M_0 - \frac{Δm}{Δt} \cdot t \]