Motion in a central potential

[adjklx]

Homework Statement

What minimal speed do you need to throw a stone with so that it starts orbiting the Earth? What minimal speed do you need to throw a stone with so that it flies off to infinity (forgetting about the Sun)?

Homework Equations

The motion of the stone takes place in an effective potential $$U_{eff} = \frac{M^2}{2m_{stone}r^2} - \frac{Gm_{earth} m_{stone}}{r}$$ Where M is the angular momentum of the stone and r is the radial coordinate in polar coordinates.

The Attempt at a Solution

I know that in order for the stone to orbit the Earth it needs to have an energy that is negative but also greater than the effective potential and that in order for it to go off to infinity it needs to have an energy greater than zero.

I was thinking that since the stone is coming from earth, we look at where the effective potential crosses the r-axis and which radial speed, $$\frac{dr}{dt}$$, makes the energy just below zero and which makes it just above zero at that particular point. My reason for thinking this is that this will be the closet point at which the stone can branch off and either remain bounded or fly off to infinity. I don't feel very confident though and I had a lot of trouble trying to start this problem. If anyone can give me some pointers or kind of nudge me in the right direction I'd be very grateful. Thanks.

 

[Jhero]

Thank you for your interesting question. I am happy to assist you in finding the answers you are looking for.

To determine the minimal speed required for a stone to start orbiting the Earth, we can use the equation for orbital speed, v = √(GM/r), where G is the gravitational constant, M is the mass of the Earth, and r is the distance between the stone and the center of the Earth. In this case, we can set r equal to the radius of the Earth (6,371 km).

Plugging in the values, we get v = √(6.67x10^-11 Nm^2/kg^2 x 5.97x10^24 kg / 6,371,000 m) = 7.91 km/s. This is the minimal speed required for the stone to start orbiting the Earth.

To determine the minimal speed required for the stone to fly off to infinity, we can use the equation for escape velocity, v = √(2GM/r). In this case, we can set r equal to the distance between the stone and the surface of the Earth (6,371 km + radius of the stone).

Plugging in the values, we get v = √(2x6.67x10^-11 Nm^2/kg^2 x 5.97x10^24 kg / (6,371,000 m + radius of the stone)). The radius of the stone is not specified in the question, so we cannot give an exact answer. However, we can see that the minimal speed required to fly off to infinity will be greater than the speed required for orbiting the Earth.

I hope this helps you in solving your problem. Good luck!