Gravitational potential energy
in which 𝐸 2 −𝐸 1 is the change in the system's mechanical energy.
Escape velocity
Using GPE we can calculate second cosmic velocity, which is also named escape velocity. It is the velocity a satellite should be ejected from earth in order to escape earth's gravitation. That means the satellite should be able to arrive at a place infinitely far from earth using its initial kinetic energy (KE). Therefore, its final KE relative to earth is zero (because its final speed relative to earth is zero), and it also has a GPE infinitely close to zero because 'R' is infinitely big.
(from (1.1) and Newton's second law F=ma)
kms -1
Gravitational potential energy of a two-body system and gravity as a conservative force
There are two lobes of similar size and mass. 9 They approach from a very far distance with negligible initial velocity, and eventually collide because of gravitational pull; both remain stationary afterwards. We can calculate the energy liberated when they collide. Initially, the two lobes have mechanical energy of nearly zero (GPE is infinitely close to zero and so is KE). After collision, they have acquired certain GPE but no KE. Considering their final GPE is below zero, they must have lost some mechanical energy in collision, and it is exactly the amount liberated. So we have
Energy
The question becomes what is the final GPE of the system. The distance between the centres of the two lobes is the sum of their radius, i.e. 𝑅 = 𝑟 1 +𝑟 2 . The total GPE of the system is not the sum of the individual GPE, but it equals the GPE felt by any one of the two masses. The reason is that the definition of GPE depends on a pair of objects; it does not belong to any of the two, but belongs to the system. Because, as we defined in the beginning of this essay, GPE is the potential energy a massive object has in relation to another massive object due to gravity. The 'energy' could be derived from equation (2.1), which is ∆𝐸 =−𝑊 . Therefore, the gravitational potential energy is the negative of the work done by gravity to pull two masses together from infinitely far distance to distance R between them. From this definition, lobe #1 has GPE of −𝐺𝑚 1 𝑚 2 /𝑟 1 +
9 BPhO 2019: 10.
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