A rocket of mass 20000 kg is ejecting the gases at the rate of 50 kg/sec with speed of 20 m/sec in gravity free space. Find the acceleration of rocket at 2 sec.

Answer

To find the acceleration of the rocket at 2 seconds, we can follow these steps:

### Step 1: Identify the given values
- Mass of the rocket (m₀) = 20,000 kg
- Rate of mass ejection (dm/dt) = 50 kg/s
- Speed of ejected gases (v) = 20 m/s

### Step 2: Calculate the thrust force (F)
The thrust force can be calculated using the formula:
\[ F = v \times \frac{dm}{dt} \]
Substituting the values:
\[ F = 20 \, \text{m/s} \times 50 \, \text{kg/s} = 1000 \, \text{N} \]

### Step 3: Calculate the mass of the rocket after 2 seconds
After 2 seconds, the mass of the rocket (m) can be calculated as:
\[ m = m₀ - \left(\frac{dm}{dt} \times t\right) \]
Substituting the values:
\[ m = 20,000 \, \text{kg} - (50 \, \text{kg/s} \times 2 \, \text{s}) \]
\[ m = 20,000 \, \text{kg} - 100 \, \text{kg} = 19,900 \, \text{kg} \]

### Step 4: Calculate the acceleration (a) using Newton's second law
According to Newton's second law:
\[ F = m \times a \]
We can rearrange this to find acceleration:
\[ a = \frac{F}{m} \]
Substituting the values:
\[ a = \frac{1000 \, \text{N}}{19,900 \, \text{kg}} \]

### Step 5: Calculate the final value of acceleration
Calculating the above expression:
\[ a \approx 0.05025 \, \text{m/s}^2 \]

Thus, the acceleration of the rocket at 2 seconds is approximately:
\[ a \approx 0.05025 \, \text{m/s}^2 \]

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