Planetary Motion: Elliptic Equation, Venus Orbit, Satellite Velocity

[fahd]

hi i hv these 2 doubts...i tried using the elliptic equation:
r= a(1-e^2)/[1-e cos(theta)] however i can't really figure out how do i determine the answer,,i mean if i could know what exactly to do.

1) the orbit of venus is very near circular (e=0.0068).Assuming that the orbit is completely circular, what wud be the orbital motion of Venus if the mass of the sun were to suddenly drop by a factor of two?Would Venus be able to remain in the solar sysytem?if so at what radius cud it be found?

2)A satellite moves in an elliptic orbit with e= 0.5 around a planet from which it is launched.When it arrives at an apsis (a radial turning point),its velocity is suddenyl doubled .Show that the new orbit will be either paraboloc or hyperbolic according to which of the turning points the velocity doubling occurs?


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i really want to know how to do these so that i can apply similar concepts to solve the other questions too..
also another quick question...given 'r' describing the motion of a planet in ,how can i find the speed?
thanks.

 

[QuantumDefect]

The first one you start with Conservation of energy. You can find the kinetic energy by using Centripetal acceleration. Once you know that, you plug it into your Cons. of Energy. Equation. Then you look look at the potential energy term. You know that if the mass of the sun drops suddenly to half of its origional value, then the potential energy changes specifically M_sun ---> .5M_Sun. Youll find that the total energy goes to zero! And because its zero, it goes into a parabolic orbit.
You can find the speed at a particular point in the orbit by using conservation of angular momentum if you have a velocity at some point. Or you could differentiate with respect to time, your orbit equation
As for your next question ill have to think about it for awhile. Hope this helps

 

[emptymaximum]

they can both be solved in the same manner. when dealing with conic section remember that the eccentricity $\epsilon$ is what determines the form of the conic. If $\epsilon \geq 1$ then the orbit will not be a closed path. recall:
$$\epsilon = \sqrt{1 + E \frac{2 m { \ell }^2} {k^2}}$$
where k is the constant from the potential energy function (2GMm), and $\ell = | \vec{r} \times \vec{v} |$

also, in your problem two, an apsis isn't a 'turning point' as the particle does not reverse motion.
an apsis is where r has a maximum/minimum value, which correspong to the perihilion and aphelion.

 

[fahd]

thanks

thanks a lot for all ur help...
ill make sure i apply this concept to the other questions as well..
thanks again!

 

[fahd]

thanks a lot for all ur help...
ill make sure i apply this concept to the other questions as well..
thanks again!

a quick question..dun u think. the constant K shud be GMm instead of
2GMm.
thanks

 

[emptymaximum]

yup.
don't know why i put a 2 there.
sorry.
/s

 

[fahd]

2)A satellite moves in an elliptic orbit with e= 0.5 around a planet from which it is launched.When it arrives at an apsis (a radial turning point),its velocity is suddenyl doubled .Show that the new orbit will be either paraboloc or hyperbolic according to which of the turning points the velocity doubling occurs?

i am not quite knowing homework to do this question..tried all stuff..but always contradicts facts..i used the equation for mechanicsla energy conservation and then tried to state that fopr parabolic or hyperbolic motion, e has to be =0 and >0 respectively...however for the parabolic path its ok but its contradicting my answer for the hperbolic type
can sumone help with sum other method??
thanks...:(

 

[emptymaximum]

no
$\epsilon = 1$ for parabolic orbit, and > 1 for hyperbolic
when $\epsilon = 0$ , it is circular orbit and when $$0 < \epsilon < 1$$ it is elliptical.