A rocket of mass `40kg` has `160kg` fuel. The exhaust velocity of the fuel is `2.0km//s`. The rate of consumption of fuel is `4kg//s`. Calculate the ultimate vertical speed gained by the rocket. `(g=10m//s^2)`

Options

A. 2km/s
B. 2.82km/s
C. 1km/s
D. 10km/s

Explanation

To solve the problem of calculating the ultimate vertical speed gained by the rocket, we can follow these steps:

### Step 1: Identify the given data
- Mass of the rocket (m) = 40 kg
- Mass of the fuel = 160 kg
- Exhaust velocity of the fuel (v_r) = 2.0 km/s = 2000 m/s
- Rate of consumption of fuel (dm/dt) = 4 kg/s
- Acceleration due to gravity (g) = 10 m/s²

### Step 2: Calculate the initial mass of the rocket (m₀)
The initial mass of the rocket when it is fully fueled is the sum of the mass of the rocket and the mass of the fuel:
\[ m_0 = m + \text{mass of fuel} = 40 \, \text{kg} + 160 \, \text{kg} = 200 \, \text{kg} \]

### Step 3: Calculate the time until all the fuel is consumed
To find out how long it will take for the fuel to be consumed, we divide the total mass of the fuel by the rate of consumption:
\[ \text{Time} (t) = \frac{\text{mass of fuel}}{\text{rate of consumption}} = \frac{160 \, \text{kg}}{4 \, \text{kg/s}} = 40 \, \text{s} \]

### Step 4: Use the formula for ultimate vertical speed
The formula for the ultimate vertical speed \( v \) of the rocket is given by:
\[ v = -g t + v_r \ln\left(\frac{m_0}{m}\right) \]
Where:
- \( m \) is the final mass of the rocket after all the fuel has been consumed, which is 40 kg.
- \( m_0 \) is the initial mass of the rocket, which we calculated as 200 kg.

### Step 5: Substitute the values into the formula
Now we can substitute the values into the formula:
1. Calculate \( \ln\left(\frac{m_0}{m}\right) \):
\[ \frac{m_0}{m} = \frac{200 \, \text{kg}}{40 \, \text{kg}} = 5 \]
Therefore, \( \ln(5) \approx 1.609 \).

2. Substitute into the formula:
\[ v = -10 \times 40 + 2000 \times 1.609 \]
\[ v = -400 + 3218 \]
\[ v = 2818 \, \text{m/s} \]

### Step 6: Convert to km/s
Finally, to convert the speed from meters per second to kilometers per second:
\[ v = \frac{2818}{1000} = 2.818 \, \text{km/s} \]

### Final Answer
The ultimate vertical speed gained by the rocket is approximately **2.818 km/s**.

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