The initial velocity vᵢ required to project a body vertically upward from the surface of the earth to reach a height of 10R, where R is the radius of the earth, may be described in terms of escape velocity vₑ such that vᵢ = √(x/y) × vₑ. The value of x will be ____________.

Solution

Physics sir · 4 steps

Height 10R means the body ends 11R from the centre. Conserve energy between R and 11R.

1. Find final distance. Gravitational potential energy depends on distance from the centre of the earth. A height of 10R above the surface puts the body at r = R + 10R = 11R, where it momentarily stops.

   1/2mvᵢ² - GMm/R = 0 - GMm/11R

2. Solve for vi. This gives vᵢ² = 2GM/R(1 - 1/11) = 2GM/R × 10/11.
3. Bring in escape velocity. Escape velocity satisfies vₑ² = 2GM/R. So vᵢ² = 10/11vₑ², that is vᵢ = √(10/11) × vₑ.
4. So. Comparing with √(x/y) × vₑ gives x = 10 and y = 11. The answer is 10.

The common slip: 9. Taking the final distance as 10R instead of 11R gives vᵢ = √(9/10) × vₑ, so x = 9. The height is measured from the surface, so add R.

The idea to keep. In gravitation, energy always uses distance from the centre, r = R + h. Write vₑ² = 2GM/R and the speed for any height comes out as a fraction of vₑ.

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