Confirming: Total & Kinetic Energy of an Object in Orbit

[decamij]

Just want to confirm. If an object is in orbit,, around earth, for example, the its total energy is equal to half of its Eg. But what is its kinetic energy equal to? Is there another expression rather than Ek = 1/2mv^2?

 

[da_willem]

For circular motion:

$$E_k = \frac{1}{2}m\frac{(2 \pi r)^2}{T^2}$$

With T the period of the orbital motion. Now if you equate the gravitational force to the centripetal force $mv^2/r$ and use the same expression as above for the velocity you get:

$$T^2=\frac{(2 \pi r)^2 r}{GM}$$

which is Keplers law for circular orbits. Filling this expression in the kinetic energy expression:

$$E_k = \frac{GMm}{2r}=-\frac{E_g}{2}$$

This shouldn't be too surprising as total energy is kinetic + potential energy and you already noticed this equalled half the potential energy.
$$E_{pot}=E_g=-G\frac{Mm}{r}$$

 

[wais]

Yes, you are correct in that the total energy of an object in orbit is equal to half of its gravitational potential energy (Eg). This is known as the conservation of energy principle, where the total energy of a system remains constant.

The kinetic energy (Ek) of an object in orbit can be calculated using the formula Ek = 1/2mv^2, where m is the mass of the object and v is its velocity. This equation represents the energy that an object possesses due to its motion.

There is no other expression for kinetic energy in this scenario as it is a fundamental concept in physics. However, it is important to note that the velocity in this equation is the orbital velocity, which is different from the linear velocity of the object. The orbital velocity takes into account the circular motion of the object around the Earth.

I hope this helps to clarify your understanding. Keep up the good work!