Congratulations for Brown getting one entry in the table right given everything else he writes is so comprehensively wrong.
When I did my own calculations the KE (kinetic energy) was based on the escape velocity of objects from Earth 11.2 km/s which represented the minimum KE required.
When considering where these objects end up in the solar system requires extra calculations such as escape velocity relative to the Sun and whether the direction of motion was the same as the Earth’s orbit (prograde) or opposite (retrograde).
In the case of comets in the Oort cloud the gravitational force is so weak, the orbits are easily perturbed by external forces such as passing stars and molecular clouds.
In this case in Brown’s scenario, a comet launched from Earth radially, the
additional tangential velocity component in the prograde direction required is √(42.1² - 29.78²) ≈ 29.8 km/s where 42.1 and 29.78 are the escape velocity from the Sun at 1 AU (astronomical unit) and the Earth’s orbital speed respectively.
The launch velocity from Earth in the prograde direction Vp = √(11.2² + 29.8²) ≈ 31.8 km/s agreeing with Brown’s result when rounded.
A comet launched from Earth in the retrograde direction which Brown has ignored, the
additional velocity is simply the vector addition 42.1 + 29.78 ≈ 71.9 km/s.
The launch velocity from Earth in the retrograde direction Vr = √(11.2² + 71.9²) ≈ 72.7 km/s.
For objects closer to the Sun the calculations are more complicated as the resulting orbits are more stable and the escape velocities are such that the objects remain in these orbits.
This is known as the Hohmann Transfer orbit.
As an example consider a TNO at an orbit of 50 AU transferred from an Earth orbit at 1 AU.
The semi major axis (a) for a transfer is a = (1 + 50)/2 = 25.5 AU.
The velocity v at the Earth’s orbit is v = √(GMₛ(2/rₑ - 1/a)) ≈ 41.7 km/s.
The velocity relative to Earth vₑ = 41.7 - 29.78 ≈ 11.9 km/s.
The launch velocity Vp in the prograde direction is Vp = √(11.2² + 11.9²) ≈ 16.4 km/s.
The velocity vₑ in the retrograde direction is vₑ = 41.7 + 29.78 = 71.48 km/s.
The launch velocity in the retrograde direction Vr = √(11.2² + 71.48²) ≈ 72.3 km/s.
The corrected table is shown as follows:
There should be a contest that anyone who can find the greatest number of errors in Hydroplate theory is deserving of a prize.