- #1
mcranfo
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Hello! I'll pose the question first:
A rocket is fired straight up, burning fuel at the constant rate of b kilograms per second. Let v=v(t) be the velocity of the rocket at time t and suppose that the velocity u of the exhaust gas is constant. Let M=M(t) be the mass of the rocket at time t and note that M decreases as the fuel burns. If we neglect air resistance, it follows from Newton’s Second Law that:
F= M (dv/dt)-ub where the force F=-Mg. Thus,
Equation I: M(dv/dt) - ub = -Mg.
Let M1 be the mass of the rocket without fuel, M2 the initial mass of the fuel
a) Find an equation for the mass M at time t in terms of M1, M2, and b.
b) Substitute this expression for M into equation 1 above and solve the resulting equation for dv/dt using separation of variables.
c) Determine the velocity of the rocket at the burnout velocity, when all the fuel is exhausted.
d) Find the height of the rocket at the burnout time.
For part a, I determined that M would equal M1 + M2 - bt. That makes sense.
For part b, I don't really know what they mean solve for dv/dt using separation of variables, so I just plugged it into equation 1 and, by solving for dv/dt, got:
(dv/dt) = -g + (ub / (M1 + M2 - bt))
So I'm not sure if I did part b correctly.
And for part c, I would assume that you would just integrate what you get in part b. What would that come out to be? I'm not very good at integration... :/
And for part d, the height of the rocket at the burnout time would be when... I'm not sure of that either...
A rocket is fired straight up, burning fuel at the constant rate of b kilograms per second. Let v=v(t) be the velocity of the rocket at time t and suppose that the velocity u of the exhaust gas is constant. Let M=M(t) be the mass of the rocket at time t and note that M decreases as the fuel burns. If we neglect air resistance, it follows from Newton’s Second Law that:
F= M (dv/dt)-ub where the force F=-Mg. Thus,
Equation I: M(dv/dt) - ub = -Mg.
Let M1 be the mass of the rocket without fuel, M2 the initial mass of the fuel
a) Find an equation for the mass M at time t in terms of M1, M2, and b.
b) Substitute this expression for M into equation 1 above and solve the resulting equation for dv/dt using separation of variables.
c) Determine the velocity of the rocket at the burnout velocity, when all the fuel is exhausted.
d) Find the height of the rocket at the burnout time.
For part a, I determined that M would equal M1 + M2 - bt. That makes sense.
For part b, I don't really know what they mean solve for dv/dt using separation of variables, so I just plugged it into equation 1 and, by solving for dv/dt, got:
(dv/dt) = -g + (ub / (M1 + M2 - bt))
So I'm not sure if I did part b correctly.
And for part c, I would assume that you would just integrate what you get in part b. What would that come out to be? I'm not very good at integration... :/
And for part d, the height of the rocket at the burnout time would be when... I'm not sure of that either...