The radius in kilometer to which the present radius of earth (R = 6400 km) to be compressed so that the escape velocity is increased 10 times is ___________.

Solution

Physics sir · 4 steps

Mass stays the same when you squeeze the earth, so escape speed goes as one over root R.

1. Escape velocity. For a planet of mass M and radius R, the escape velocity is given below. Compressing the earth does not change its mass, so GM is a constant.

   vₑ = √(2GM/R) ⇒ vₑ ∝ 1/√R

2. Take the ratio. vₑ'/vₑ = √(R/R') = 10. Squaring both sides gives R/R' = 100.
3. Put in numbers. R' = R/100 = (6400 km)/100 = 64 km.
4. So. The earth must be compressed to a radius of 64 km.

The common slip: 640. Treating vₑ as proportional to 1/R and dividing the radius by 10 gives 640 km. The square root means you must divide by 10² = 100.

The idea to keep. Check what stays fixed first. With mass fixed, vₑ ∝ 1/√R, so to multiply vₑ by n you must divide R by n².

Answer from the NTA final answer key. Solution by PrepWiser.