Map d Sapling Learning macmillan learning A rocket is launched vertically up wit
ID: 1792491 • Letter: M
Question
Map d Sapling Learning macmillan learning A rocket is launched vertically up with no initial velocity. Propulsion is provided by the ejection of mass with constant velocity of ejection u = 32.0 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 50.0% of the total mass at launch. Assuming constant gravitational acceleration, how long does it take the rocket to achieve maximum upward acceleration? Number Previous Give Up & View Solution O Check Answer Next Exit HintExplanation / Answer
If we assume the initial mass is M, then the initial weight of the rocket is
W = g m.
Since the initial acceleration is zero, this weight is countered by thrust.
T = g m
T * t = gm t
dp = gm t
m(exhaust) * dv = gm(rocket) t
m(exhaust) = g m(rocket) t/ dv
m(exhaust/rocket) = g t / dv
m(exhaust/rocket) = 9.81m/s * 1s / 32 m/s
m(exhaust/rocket) = 0.306
So 0.306 kg/s ejected at 32m/s will provide a force of 9.81N, sufficient to support 1kg of mass. This means the rocket begins with a rate of 0.306 rockets/s, from m=1 to m=0.19.
dm = 0.50m
rate = 0.306m(rocket)/s
t = dm / rate
t = 0.50m / 0.306m(rocket)/s
t = 1.63 s
So rocket will be accelerating during the launch until fuel cut-off. That will happen at 1.63 s and will be the point of maximum acceleration.