A rocket is launched vertically up with no initial velocity. Propulsion is provi

Question

A rocket is launched vertically up with no initial velocity. Propulsion is provided by the ejection of mass with constant velocity of ejection u = 49.0 m/s relative to the rocket and at a constant rate so determined that the initial acceleration is zero. The mass of fuel that can be ejected is 66.0% of the total mass at launch. Assuming constant gravitational acceleration, how long does it take the rocket to achieve maximum upward acceleration?

Hint:
Express the momentum-impulse theorem in the presence of the gravitational force and find the time depence of the rocket's mass using the initial conditions (zero initial acceleration) and that the mass is ejected at constant rate. where ?p is the change in momentum of the rocket and FG = mg is the gravitational force. Substituting the expression found back into the impulse-momentum theorem, the time dependence of the acceleration can be found. From there, it can be determined how acceleration changes in time. The time at which acceleration is maximum should be before all the fuel was ejected.

Explanation / Answer

Suppose mass is ejected at x kg/s.
That gives the rocket a change in momentum, ie a force, of 49x kg m/s/s.
The initial acceleration is zero, so this force equals the gravitational force Mg.
x = 0.158 M kg/s

The constant force will provide more and more acceleration as the mass of the rocket decreases.
The maximum upward acceleration occurs right before the rocket runs out of fuel.
0.66 M / x = 0.23 s

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