Mechanics with increasing mass and decreasing g. (1 Viewer)

Drsoccerball · Feb 25, 2016

How do we solve mechanics questions where mass changes and also g changes. For example a rocket entering a space cloud the g increases the closer you get and the mass of the rocket increases. Also shouldn't we work with variable g when doing any mechanics question in the vertical motion? How do we do these questions, are they examples of differential equations?

Paradoxica · Feb 25, 2016

To sum it up nicely: The complexity of the mathematics involved is exponentially proportional to the number of dimensions/variables/whatever you have.

InteGrand · Feb 25, 2016

Drsoccerball said:
How do we solve mechanics questions where mass changes and also g changes. For example a rocket entering a space cloud the g increases the closer you get and the mass of the rocket increases. Also shouldn't we work with variable g when doing any mechanics question in the vertical motion? How do we do these questions, are they examples of differential equations?

$\noindent Let the mass as a function of time be $M=M(t)$ (assume it is a smooth function). Then $\bold{F}_G = M\bold{a}_G$, where $\bold{a}_G$ is the acceleration due to gravity (which depends on position, and hence also time), and $\bold{F}_G$ is the force due to gravity on the object (again depends on space, and time). Using Newton's second law, $\bold{F}_\text{net} = M \frac{\mathrm{d} \bold{v}}{\mathrm{d}t}+\bold{v}\frac{\mathrm{d}M}{\mathrm{d}t}$, where $\bold{v}$ is the velocity of the object and $\bold{F}_\text{net}$ is the net force on it. If we are working in 1D, these vectors can be treated like scalars due to uses of signs if necessary.$

$\noindent Now, if the only force on the object is due to gravity, the differential equation we get is: $M\bold{a}_G =M \frac{\mathrm{d} \bold{v} }{\mathrm{d}t} + \bold{v} \frac{\mathrm{d}M}{\mathrm{d}t}$.$

$\noindent Did you have any particular example, where $M$ is given as a function of time, and we are told what the forces on the object are? If so, we can formulate a specific differential equation.$

Nailgun · Feb 26, 2016

Drsoccerball said:
How do we solve mechanics questions where mass changes and also g changes. For example a rocket entering a space cloud the g increases the closer you get and the mass of the rocket increases. Also shouldn't we work with variable g when doing any mechanics question in the vertical motion? How do we do these questions, are they examples of differential equations?

Oooh oooh, I asked a question about this last year when doing Space.
My teacher ended up getting one of that maths teachers to derive the forumula for it so I'm fairly sure he used mechanics
It was before I learned calculus though, so I didn't understand it at the time

actually lol can anyone like, do it for me
Basically, an equation for the velocity of a rocket moving away from the Earth
$$Due to Conservation of Momentum, we see that as the mass of a rocket decreases (as fuel is consumed) the velocity of the rocket increases. A different analysis is that from Newton's Second Law $ F=ma $ we see that if the thrust force acting on the rocket remains constant, as the mass decreases so does the acceleration increase, implying a non-uniform acceleration to the rocket. The other thing to take into account is that according to $ F_{ G }=\frac { G{ m }_{ 1 }m_{ 2 } }{ { d }^{ 2 } } $ As the distance from the planet increases, the force acting on the rocket decreases at a decreasing rate. Can someone resolve all of these lol$

At the very least I'm pretty sure it's possible (since the maths teacher did it)

InteGrand · Feb 26, 2016

Nailgun said:
Oooh oooh, I asked a question about this last year when doing Space.
My teacher ended up getting one of that maths teachers to derive the forumula for it so I'm fairly sure he used mechanics
It was before I learned calculus though, so I didn't understand it at the time

actually lol can anyone like, do it for me
Basically, an equation for the velocity of a rocket moving away from the Earth
$$Due to Conservation of Momentum, we see that as the mass of a rocket decreases (as fuel is consumed) the velocity of the rocket increases. A different analysis is that from Newton's Second Law $ F=ma $ we see that if the thrust force acting on the rocket remains constant, as the mass decreases so does the acceleration increase, implying a non-uniform acceleration to the rocket. The other thing to take into account is that according to $ F_{ G }=\frac { G{ m }_{ 1 }m_{ 2 } }{ { d }^{ 2 } } $ As the distance from the planet increases, the force acting on the rocket decreases at a decreasing rate. Can someone resolve all of these lol$

At the very least I'm pretty sure it's possible (since the maths teacher did it)

I think I did something similar here for constant gravity: http://community.boredofstudies.org/18/physics/344618/hsc-physics-marathon-2016-a-4.html#post7069436

$\noindent For varying gravity, the differential equation is harder to solve if I recall correctly (I think I thought about this before and didn't know how to solve it, if indeed it is possible to solve (not all differential equations can be solved analytically)), maybe I'll have a look at it later. (To obtain the differential equation, replace the $g$ in my linked post with $\frac{GM(t)m_E}{x^2}$ in the line where I obtained the differential equation for constant $g$, where $m_E$ is the mass of the Earth, $x$ is displacement, $G$ is obvious, and $M(t)$ is what it is in that post. The resulting ODE is non-linear and second order, and there's generally little known about these types of ODE's. (ODE means ordinary differential equation.)$

InteGrand · Feb 26, 2016

If we ignore all external forces though (so ignore gravity), there is this result: https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation .

Mechanics with increasing mass and decreasing g. (1 Viewer)

Drsoccerball

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Paradoxica

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InteGrand

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Nailgun

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InteGrand

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InteGrand

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