-with a given amount of fuel, a rocket in space can change (increase) its velocity by a certain amount regardless of how fast it is already moving, so suppose it can change its velocity by 5km/s
-kinetic energy increases at the square of velocity
-the formula for kinetic energy is KE (in joules) = 1/2 * (Mass KG)* (Velocity m/s)^2
so if we have the passenger section weighing 10000kg of a rocket thats already going 250km/s, the kinetic energy of the passenger section is:
1/2 * (10000kg) * (250000m/s)^2 = 3.125*10^14 joules
if the rocket increases its velocity by 5km/s, the kinetic energy is now:
1/2 * (10000kg) * (255000m/s)^2 = 3.25125*10^14 joules
but suppose the rocket which was initially 250km/s is falling towards a black hole, and at close approach reaches 32301km/s, its kinetic energy is now:
1/2 * (10000kg) * (32301000m/s)^2 = 5.216773005 * 10^18 joules
suppose the rocket has waited to do its 5km/s burn until this time, it’s kinetic energy is now:
1/2 * (10000kg) * (32306000m/s)^2 = 5.21838818 * 10^18 joules
if the rocket hadn’t done a burn, and coasted to a point where it slowed back to 250km/s it would have lost this amount of kinetic energy since close approach:
(1/2 * (10000kg) * (32301000m/s)^2) - (1/2 * (10000kg) * (250000m/s)^2) = 5.216460505*10^18 joules
but if the rocket indeed does the 5kms burn at close approach, increasing its velocity from 32301km/s to 32306km/s it still will lose the same amount of kinetic energy at the same outbound distance, leaving it with this amount of kinetic energy:
(1/2 * (10000kg) * (32306000m/s)^2) - (5.216460505 * 10^18 joules) = 1.927675*10^15 joules
the remaining velocity at the same distance where the craft had 250km/s inbound (with 5km/s burn at close approach) is now 620.91km/s:
1/2 * (10000kg) * (620914.647918m/s)^2 = 1.927675 * 10^15 joules
^In other words, with 250km/s inbound, increasing from gravity to 32301km/s at close approach, increasing further to 32306km/s with a 5km/s rocket burn at close approach, and then coasting to the same distance where the craft initially had 250km/s, the craft has 620.91km/s instead. The 5km/s rocket burn at close approach caused an effective velocity change of 370.91km/s which is a multiplication of effectiveness of the burn of a factor 74.1x, thanks to an Oberth maneuver in the vicinity of the black hole.