A 6100 kg rocket is set for vertical firing from the ground....

A 6100 $\mathrm{kg}$ rocket is set for vertical firing from the ground. If the exhaust speed is $1200 \mathrm{m} / \mathrm{s},$ how much gas must be ejected each second if the thrust (a) is to equal the magnitude of the gravitational force on the rocket and (b) is to give the rocket an initial upward acceleration of 21 $\mathrm{m} / \mathrm{s}^{2} ?$

A 6100 $\mathrm{kg}$ rocket is set for vertical firing from the ground. If the exhaust speed is $1200 \mathrm{m} / \mathrm{s},$ how much gas must be
ejected each second if the thrust (a) is to equal the magnitude of
the gravitational force on the rocket and (b) is to give the rocket an
initial upward acceleration of 21 $\mathrm{m} / \mathrm{s}^{2} ?$

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Key Concepts

Mass Flow Rate in Propulsion

The mass flow rate, dm/dt, is the rate at which mass is expelled from the rocket. It is directly tied to the thrust produced by the rocket's engines and is determined by rearranging the thrust equation to dm/dt = T / v_ex. This concept is critical in calculating how much fuel or propellant must be ejected per unit time to achieve specific force requirements.

Newton's Second Law in Rocket Motion

Newton's second law states that the net force acting on an object is equal to its mass times its acceleration (F_net = m * a). In the context of rockets, this means that to achieve a desired acceleration, the total thrust must overcome both gravitational forces (mg) and provide the necessary additional force for acceleration.

Thrust in Rocket Propulsion

Thrust in rocket propulsion is produced by ejecting mass at high speed, which, according to the conservation of momentum, generates an equal and opposite force on the rocket. The relationship is given by T = (dm/dt) * v_ex, where T is thrust, dm/dt is the mass flow rate of the ejected gas, and v_ex is the exhaust velocity.

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