In this problem, we're dealing with the physics of rocket propulsion. The thrust needed to lift a rocket and provide additional acceleration depends on both the rocket's weight and the desired acceleration. To determine how much gas must be ejected each second, we'll use the principles of rocket motion described by the Tsiolkovsky rocket equation and Newton's second law. Let's break this down step by step.
Given Information
- Rocket mass (m): 5860 kg
- Exhaust speed (v_e): 1.17 km/s (convert to meters per second: 1170 m/s)
- Gravitational acceleration (g): 9.8 m/s²
- Initial upward acceleration (a): 18.3 m/s²
What We Need to Find
We need to calculate the rate of mass ejection (the mass of gas expelled per second, denoted as dm/dt) for two conditions:
- The thrust needed to overcome the weight of the rocket.
- The thrust needed to give the rocket an initial upward acceleration of 18.3 m/s².
Thrust and the Rocket Equation
The total thrust F generated by the rocket is the force needed to overcome gravity and provide the necessary acceleration. The relationship between thrust and mass ejection rate is given by:
F = v_e × (dm/dt)
Where:
- F is the thrust required (in newtons),
- v_e is the exhaust velocity of the gas (in m/s),
- dm/dt is the mass ejection rate (in kg/s).
(a) Thrust to Overcome the Weight of the Rocket
To just lift the rocket off the ground, the thrust needs to counteract the weight of the rocket. The weight W is given by:
W = m × g
Substitute the known values:
W = 5860 kg × 9.8 m/s² = 57,428 N
The thrust required to overcome the weight of the rocket is 57,428 N. Now, using the thrust equation F = v_e × (dm/dt), we can solve for the mass ejection rate:
dm/dt = F / v_e = 57,428 N / 1170 m/s ≈ 49.04 kg/s
Thus, the rocket must eject gas at a rate of approximately 49.04 kg/s to overcome its weight.
(b) Thrust to Give the Rocket an Initial Upward Acceleration
To give the rocket an upward acceleration, the thrust must also overcome its weight and provide the additional force required to accelerate the rocket. From Newton's second law, the total force required is:
F = m × (g + a)
Substituting the given values:
F = 5860 kg × (9.8 m/s² + 18.3 m/s²) = 5860 kg × 28.1 m/s² = 164,246 N
Now, we can calculate the mass ejection rate needed to generate this thrust:
dm/dt = F / v_e = 164,246 N / 1170 m/s ≈ 140.4 kg/s
Thus, to give the rocket an initial upward acceleration of 18.3 m/s², the gas must be ejected at a rate of approximately 140.4 kg/s.
Summary of Results
- Mass ejection rate to overcome weight: 49.04 kg/s
- Mass ejection rate for an upward acceleration of 18.3 m/s²: 140.4 kg/s
These values represent how much gas the rocket must expel each second to achieve the necessary thrust in both situations.
