Hello Mark,
I'll give you a very general outline of the calculation and the theory without giving a specific answer as the resulting integral is very difficult to solve. Also the derivation of the integral requires a working knowledge of the Schwarzschild metric which is a static Black Hole solution for General Relativity.
I'll leave it as an exercise to solve the integral although I suspect there is no direct solution and would need to be solved by numerical methods.
In my response to Peter I used embolden terms coordinate and proper velocity.
To solve your problem we need to look at coordinate and proper time by incorporating a third observer who is located outside the gravitational well of the black hole.
To simplify matters we will assume the object is released from rest and falls into a non rotating black hole.
This third observer will measure the coordinate time it takes for the object to be released from rest 25 km from the singularity and free falls to 5 km from the singularity.
The coordinate time is the elapsed time measurement based from the third observer's frame of reference. The object is sending a signal to this observer.
If on the other hand there is a clock attached to the moving object, this clock is measuring proper time as the clock is in the object's frame of reference.
The mathematics for a non rotating black hole is based on the Schwarzschild metric and the time taken for an object initially at rest to fall from 25 km to 5 km in the gravitational field of a one solar mass black hole is given by the integral.
http://members.iinet.net.au/~sjastro...s/freefall.gif
(attachment 1)
c is the speed of light. G is the gravitational constant, r is the radial distance.
As stated this is one very difficult integral to solve and the answer is based on the coordinate time of the distant third observer.
We will simply call it t.
The next step is to calculate the proper time T for the observers located at 100 and 10000 km from the singularity respectively.
Since these observers are located in different locations in the gravitational well their clocks will run slower compared to the coordinate time measured by the distant third observer.
One is comparing the proper times of these observers based on how slow their local clocks run against the distant observers coordinate time.
The equation in this case is much more simple.
http://members.iinet.net.au/~sjastro...e_dilation.gif
(attachment 2)
rs is the event horizon and has a value of 2.95 km.
r is the radial distance of the observers from the singularity.
t is the coordinate time for the third observer.
For the observer at 10000 km, the proper time T1 = t(1-(2.95/10000)^0.5)
For the observer at 100 km the proper time T2 = t(1-2.95/100)^0.5)
Hope this helps.
Regards
Steven