Final answer:
The takeoff acceleration of a 10,000-kg anti-ballistic missile that expels 182 kg of gas per second at an exhaust velocity of 2.33 x 10³ m/s is calculated using the rocket thrust equation, resulting in an acceleration of 42.966 m/s².
Step-by-step explanation:
To calculate the takeoff acceleration of an antiballistic missile (ABM), we use the principle of conservation of momentum, specifically the equation derived from Newton's second law as applied to rockets, Thrust = mass flow rate x exhaust velocity. The force exerted by the expelled gas is equal to the mass of gas expelled per second times the exhaust velocity. This force constitutes the thrust of the missile.
To find acceleration (a), we use the formula a = F/m, where F is the thrust and m is the mass of the missile. Therefore, a = dm/dt × ve / m, where dm/dt is the mass flow rate of the gas and ve is the exhaust velocity. For the ABM:
- Mass flow rate dm/dt = 182 kg/s (given in the question).
- Exhaust velocity ve = 2.33 × 10³ m/s.
- Mass of the missile m = 10,000 kg.
Plugging these values into the formula, we get:
a = (182 kg/s × 2.33 × 10³ m/s) / 10,000 kg
= 42.966 m/s².
The takeoff acceleration of the 10,000-kg ABM is therefore 42.966 m/s².