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Standard XI
Physics
Linear Momentum of a System of Particles
Question

A rocket has total mass $$M_{i}=360\ kg$$, including $$M_{f}=330\ kg$$ of fuel and oxidizer, In interstellar space, it starts from rest at the position $$x=0$$, turns on its engine at time $$t=0$$, and puts out exhaust with relative speed $$v_{e}=1\ 500\ m/s$$ at the constant rate $$k=2.50\ kg/s$$. The fuel will last for a burn time of $$T_{b}=M_{f}/k=330\ kg/(2.5\ kg/s)=132\ s$$.
Make a graph of the velocity of the rocket as a function of time for times running from $$0$$ to $$132\ s$$.

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Q1
A rocket has total mass $$M_{i}=360\ kg$$, including $$M_{f}=330\ kg$$ of fuel and oxidizer, In interstellar space, it starts from rest at the position $$x=0$$, turns on its engine at time $$t=0$$, and puts out exhaust with relative speed $$v_{e}=1\ 500\ m/s$$ at the constant rate $$k=2.50\ kg/s$$. The fuel will last for a burn time of $$T_{b}=M_{f}/k=330\ kg/(2.5\ kg/s)=132\ s$$.
Graph the acceleration as a function of time.
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Q2
A rocket has total mass $$M_{i}=360\ kg$$, including $$M_{f}=330\ kg$$ of fuel and oxidizer, In interstellar space, it starts from rest at the position $$x=0$$, turns on its engine at time $$t=0$$, and puts out exhaust with relative speed $$v_{e}=1\ 500\ m/s$$ at the constant rate $$k=2.50\ kg/s$$. The fuel will last for a burn time of $$T_{b}=M_{f}/k=330\ kg/(2.5\ kg/s)=132\ s$$.
Graph the position during the burn as a function of time.
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Q3
A rocket has total mass $$M_{i}=360\ kg$$, including $$M_{f}=330\ kg$$ of fuel and oxidizer, In interstellar space, it starts from rest at the position $$x=0$$, turns on its engine at time $$t=0$$, and puts out exhaust with relative speed $$v_{e}=1\ 500\ m/s$$ at the constant rate $$k=2.50\ kg/s$$. The fuel will last for a burn time of $$T_{b}=M_{f}/k=330\ kg/(2.5\ kg/s)=132\ s$$.
Show that the acceleration of the rocket is
$$a(t)=\dfrac{kv_{e}}{M_{i}-kt}$$
View Solution
Q4
A rocket has total mass $$M_{i}=360\ kg$$, including $$M_{f}=330\ kg$$ of fuel and oxidizer, In interstellar space, it starts from rest at the position $$x=0$$, turns on its engine at time $$t=0$$, and puts out exhaust with relative speed $$v_{e}=1\ 500\ m/s$$ at the constant rate $$k=2.50\ kg/s$$. The fuel will last for a burn time of $$T_{b}=M_{f}/k=330\ kg/(2.5\ kg/s)=132\ s$$.
Make a graph of the velocity of the rocket as a function of time for times running from $$0$$ to $$132\ s$$.
View Solution
Q5
A rocket has total mass $$M_{i}=360\ kg$$, including $$M_{f}=330\ kg$$ of fuel and oxidizer, In interstellar space, it starts from rest at the position $$x=0$$, turns on its engine at time $$t=0$$, and puts out exhaust with relative speed $$v_{e}=1\ 500\ m/s$$ at the constant rate $$k=2.50\ kg/s$$. The fuel will last for a burn time of $$T_{b}=M_{f}/k=330\ kg/(2.5\ kg/s)=132\ s$$.
Show that during the burn the velocity of the rocket as a function of time is given by
$$v(t)=-v_{e}\ln \left(1-\dfrac{kt}{M_{i}}\right)$$
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