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 it has ejected a certain amount of mass over a specified time period. Let's break it down step by step:

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

### Step 2: Determine the Mass Ejection Rate
The mass ejection rate is defined as the amount of mass ejected per unit time. This is given by:
\[ \text{Mass rate} = \frac{\Delta m}{\Delta t} \]
where \( \Delta m \) is the mass ejected and \( \Delta t \) is the time interval over which this mass is ejected.

### Step 3: Calculate the Total Mass Ejected Over Time \( t \)
If the rocket ejects mass at a constant rate, the total mass ejected over a time period \( t \) can be calculated as:
\[ \text{Total mass ejected} = \text{Mass rate} \times t = \left( \frac{\Delta m}{\Delta t} \right) \times t \]

### Step 4: Calculate the Remaining Mass of the Rocket
The remaining mass of the rocket after time \( t \) can be found by subtracting the total mass ejected from the initial mass:
\[ M_{\text{final}} = M_0 - \text{Total mass ejected} \]
Substituting the expression for total mass ejected, we have:
\[ M_{\text{final}} = M_0 - \left( \frac{\Delta m}{\Delta t} \times t \right) \]

### Final Expression
Thus, the mass of the rocket after time \( t \) is given by:
\[ M_{\text{final}} = M_0 - \frac{\Delta m}{\Delta t} \times t \]