V=u-gt+Vr l(m0/m)
V=ultimate vertical speed
V=0-10x40+(2x10*3)ln(200/40)
=2818m/s. =2.82km/s

Given data:
- Mass of the rocket (m) = 40 kg
- Mass of the fuel (m_fuel) = 160 kg
- Exhaust velocity of the fuel (v_e) = 2 km/s
- Rate of fuel consumption (c) = 4 kg/s

Approach:
To calculate the ultimate vertical speed gained by the rocket, we need to consider the principle of conservation of momentum. According to this principle, the total momentum before and after the rocket's propulsion should be equal.

Step 1: Calculate the initial momentum:
The initial momentum of the rocket can be calculated using the formula:
Initial momentum = Mass of the rocket x Initial velocity of the rocket

Since the rocket is initially at rest, the initial velocity of the rocket is 0. Therefore, the initial momentum is also 0.

Step 2: Calculate the final momentum:
The final momentum of the rocket can be calculated using the formula:
Final momentum = (Mass of the rocket + Mass of the fuel) x Final velocity of the rocket

We need to calculate the final velocity of the rocket, which can be determined using the formula:
Final velocity of the rocket = (Rate of fuel consumption x Exhaust velocity of the fuel) / (Mass of the rocket + Mass of the fuel)

Step 3: Calculate the ultimate vertical speed:
The ultimate vertical speed gained by the rocket can be determined by subtracting the initial velocity from the final velocity.

Calculation:
- Initial momentum = 0
- Final velocity of the rocket = (4 kg/s x 2 km/s) / (40 kg + 160 kg) = 0.02 km/s
- Ultimate vertical speed = Final velocity of the rocket - Initial velocity = 0.02 km/s - 0 km/s = 0.02 km/s

Answer:
The ultimate vertical speed gained by the rocket is 0.02 km/s.