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Physics Challenge Question 12: Solutions Part 1 From class, we know that the total mechanical energy is conserved. That means that ME initial ME final . So, we need to identify what the energy is just when the object is launched off the planet, and when it is infinitely far from the planet. 1 1 2 2 KEi mvi mve (The initial velocity is the escape velocity.) 2 2 1 2 KE f mv f 0 (If it just escapes the planet, its final speed is 0.) 2 PE i G PE f G M m R M m 0 (When it is infinitely far away, r f and PE f 0 .) rf By energy conservation, we get KEi PEi KE f PE f 1 M m 2 mve G 0 0. 2 R Now, we can rewrite to solve for ve: 1 M m 2 mve G 2 R 1 2 M ve G 2 R M 2 ve 2 G R 2GM ve R Part 2 Plugging in to our equation, we get: 2 6.67 10 11 kgms 2 6.0 10 24 kg 3 ve 2 6,360,000m 1.12 10 4 m s 11.2 kms Part 3 For this part, we must rewrite our equation. (We want to solve for R.) 2GM R 2 ve Now, we can use the numbers given to estimate the radius of the black hole: 3 2 6.67 10 11 kgms 2 2.0 10 30 kg R 2,964m 3.0km 3.0 108 ms 2 Even though this is just an estimate (our escape velocity equation isn’t entirely correct for light beams), it still shows just how extremely compact black holes are: The mass of our sun (about 700,000 km in radius) has been squeezed into a radius of just 3 km!