Differentiating acceleration using constant of motion

[grav-universe]

Let's say we have an object falling through an accelerated field from r to s. I would say a gravitational field, as it is by the same process as gravity, but I will be applying a different constant of motion within the field.

If the constant of motion is something like (1 - b / r) = (1 - (v_r/c)^2), or (1 - b / r) / (1 - (v_r/c)^2) = K (constant for all r), where b is also constant, then the acceleration in terms of r can be found with

b / r = (v_r/c)^2, b / s = (v_s/c)^2

a = d(v^2) / (2 dr)

a = (v_s^2 - v_r^2) / (2 dr), where r - s = dr

a = c^2 (b / s - b / r) / (2 (r - s))

a = c^2 b (r - s) / (2 r s (r - s))

a = c^2 b / (2 r^2)

But what I want to know is how we would find the acceleration similarly to the above example with a constant of motion within an accelerated field of (1 - b / r) = (1 - v_r / c) / (1 + v_r / c) instead?

 

[jvicens]

I would first clarify the terminology being used in this scenario. In physics, an accelerated field typically refers to a field where objects experience a force that causes them to accelerate. In this case, it seems that the term "accelerated field" is being used to describe a field with a changing gravitational constant.

In order to find the acceleration in this scenario, we would need to use the equations of motion in a changing gravitational field. This can be done by using the equation:

F = m (dv/dt) = m (d^2r/dt^2)

where F is the force, m is the mass of the object, v is the velocity, and r is the position.

In this case, the force experienced by the object would be given by:

F = G (1 - b/r)^2 m_1 m_2 / r^2

where G is the gravitational constant, b is the constant in the changing gravitational field, and m_1 and m_2 are the masses of the object and the source of the gravitational field, respectively.

Using the equation F = m (d^2r/dt^2), we can then solve for the acceleration:

a = (d^2r/dt^2) = G (1 - b/r)^2 m_2 / r^2

We can then use the chain rule to express this acceleration in terms of r:

a = (dr/dt) (d^2r/dr^2) = v (dv/dr)

Finally, we can substitute in the expression for v (dv/dr) to get the acceleration in terms of r:

a = (d/dr) (v^2) = (d/dr) [(c^2 (1 - b/r)^2) / (1 - v_r/c)^2]

This expression can then be simplified and solved for the acceleration in terms of r. However, it is important to note that this is a simplified model and may not accurately reflect the true behavior of objects in a changing gravitational field. Further research and experimentation would be necessary to fully understand and accurately model this scenario.