Do you know how much energy is required to accelerate a mass 1 kg to 7 000 m/s speed? Right 24.5 MJ!
Can you do the math for a starting mass of 1 kg at 0m/s and a final mass of 0.5 kg at 7000 m/s considering a constant rate of mass loss ?
And by the way, what would be the final speed and energy of a rocket weighting 1000 kg , including 500 kg of fuel, burning it with an ejection speed of 4500 m/s ? (commmon value for oxygen/hydrogen rocket engines)
That doesn't matter so much as the shuttle didn't reduce it's speed to 0 m/s for rentry.
Assuming a fully fuel laden shuttle massing around 585,000 kg and carrying 103,000 kg of LH
2 in a low earth orbit at 7,800 m/s. H
2 has a combustion energy value of 141,000 KJ/kg. The shuttle needs to reduce it's orbital velocity from 7,800 m/s to around 4,700 m/s for re-entry.
So:
Energy required for de-orbit = 1/2 x 585,000 x (7,800-4700)
2 = 2.81 TJ
Energy available = 103,000 x 141,000 = 14.5 TJ
That's enough energy for 5 de-orbit burns. Even if you reduce the fuel to 25% of capacity it would still amount to over 3 TJ of energy, still enough to deorbit without even correcting the mass of the shuttle.