Q4. How much energy must be given to a 1000 kg spacecraft at rest on the Earth’s surface for it to escape, ignoring air resistance and the Earth’s rotation? Take g = 9.8 m s and RE = 6.4 × 10 m.
The least kinetic energy is GMEm/RE = mgRE = 1000 × 9.8 × 6.4 × 10, about 6.27 × 10 J.
Q5. To what radius would the Earth (mass 6.0 × 10 kg) have to be squeezed for its escape speed to equal the speed of light, 3.0 × 10 m s? Take G = 6.67 × 10 N m kg.
Setting √(2GM/R) = c gives R = 2GM/c = 2 × 6.67 × 10 × 6.0 × 10/(9.0 × 10), about 8.9 × 10 m, under a centimetre. (This Newtonian calculation is only a formal estimate: it is not valid for motion at light speed. It happens to match the Schwarzschild radius of a non-rotating, uncharged black hole in general relativity.)
Q6. A body is thrown straight up from the Earth’s surface at 8.0 km/s. Ignoring air resistance, how high does it rise? Take ve = 11.2 km/s, RE = 6400 km and g = 9.8 m s.
h = REv/(ve2 − v) = 6400 × 64/(125.44 − 64) = 6400 × 64/61.44, about 6670 km, slightly more than the Earth’s radius. The constant-g formula v/(2g) would give only about 3270 km.