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)`

Explanation

To solve the problem of calculating the ultimate vertical speed gained by the rocket, we can use the Tsiolkovsky rocket equation, which relates the change in velocity of a rocket to the effective exhaust velocity and the mass of the rocket and its fuel.

### Step-by-Step Solution:

1. **Identify the given values:**
- Mass of the rocket (m₀) = 40 kg
- Mass of the fuel (m_f) = 160 kg
- Total initial mass (m₀ + m_f) = 40 kg + 160 kg = 200 kg
- Exhaust velocity (v_e) = 2.0 km/s = 2000 m/s (convert km/s to m/s)
- Rate of fuel consumption (ṁ) = 4 kg/s
- Acceleration due to gravity (g) = 10 m/s²

2. **Calculate the effective mass of the rocket after fuel is burned:**
- The total mass of the rocket when fully fueled is 200 kg.
- The mass of the rocket after burning all the fuel is:
\[ m_f = 40 \text{ kg (rocket)} + 160 \text{ kg (fuel)} - 160 \text{ kg (burned fuel)} = 40 \text{ kg} \]

3. **Use the Tsiolkovsky rocket equation:**
The equation is given by:
\[ \Delta v = v_e \ln\left(\frac{m_0}{m_f}\right) \]
where:
- \( \Delta v \) = change in velocity
- \( m_0 \) = initial total mass (rocket + fuel)
- \( m_f \) = final mass (rocket after all fuel is burned)

Substituting the values:
\[ \Delta v = 2000 \ln\left(\frac{200}{40}\right) \]

4. **Calculate the logarithm:**
\[ \frac{200}{40} = 5 \]
Therefore,
\[ \Delta v = 2000 \ln(5) \]

5. **Calculate \( \ln(5) \):**
The natural logarithm of 5 is approximately 1.609.
\[ \Delta v = 2000 \times 1.609 \approx 3218 \text{ m/s} \]

6. **Account for the effect of gravity:**
The ultimate vertical speed gained by the rocket must also consider the gravitational pull. The effective speed gained by the rocket is:
\[ v_{\text{final}} = \Delta v - \text{(time of fuel burn)} \times g \]
However, since we are calculating the ultimate speed, we consider the speed gained just before the fuel runs out.

7. **Final Result:**
The ultimate vertical speed gained by the rocket is approximately:
\[ v_{\text{final}} \approx 3218 \text{ m/s} \]