Final answer:
The takeoff acceleration of a 10,000-kg ABM that expels 220 kg of gas per second at an exhaust velocity of 2540 m/s is 56.48 m/s². The closest answer choice provided, which matches our calculation, is Option B (56.48 m/s²).
Step-by-step explanation:
To calculate the takeoff acceleration of an antiballistic missile (ABM), you can use Newton's second law of motion that states the force is equal to mass times acceleration (F = ma). According to this law and the conservation of momentum, the thrust (force) produced by the missile can be calculated as the product of the mass flow rate of the expelled gas and its exhaust velocity. This thrust can be represented by the equation F = (dm/dt) * ve, where dm/dt is the mass flow rate and ve is the exhaust velocity.
To find the acceleration (a), we rearrange the equation to a = F/m. Substituting in the given values for the mass of the missile (10,000 kg), the mass of gas expelled per second (220 kg/s), and the exhaust velocity (2540 m/s), we get:
a = ((220 kg/s) * (2540 m/s)) / (10,000 kg) = 564,800 N / 10,000 kg = 56.48 m/s2
The closest answer choice to this calculated acceleration is Option B, which is 56.48 m/s2.