Final answer:
To achieve an initial upward acceleration of 2 g, a 5000 kg rocket requires a rate of ejection of gas at 50 kg/s with an exhaust speed of 1000 m/s.
Step-by-step explanation:
The rate of ejection of gas required to provide the necessary thrust for a 5000 kg rocket to achieve an initial upward acceleration of 2 g (where g is the acceleration due to gravity on Earth, 9.8 m/s2) can be found using Newton's second law and the equation for thrust in rocketry. The total acceleration a is given by a = arocket + g, where arocket is the additional acceleration provided by the rocket's engines.
To find the thrust F we use F = m * a, where m is the mass of the rocket. Then, using the rocket thrust equation F = ve * dm/dt, where ve is the exhaust velocity and dm/dt is the rate of mass ejection, we can solve for dm/dt:
F = m * a = ve * dm/dt
dm/dt = m * a / ve
Given:
- Mass of the rocket (m) = 5000 kg
- Acceleration due to gravity (g) = 9.8 m/s2
- Desired total acceleration (2g) = 2 * 9.8 m/s2 = 19.6 m/s2
- Resulting acceleration for the additional thrust (arocket) = 19.6 m/s2 - 9.8 m/s2 = 9.8 m/s2
- Exhaust speed (ve) = 1000 m/s
Now, calculating the rate of mass ejection:
dm/dt = m * arocket / ve = 5000 kg * 9.8 m/s2 / 1000 m/s = 49 kg/s
Therefore, the closest option to 49 kg/s is 50 kg/s (option a).