A body of mass m is projected with velocity  nve in vertically upward direction from the surface of the earth into space. It is given that ve is escape velocity and n < 1. If air resistance is considered to the negligible, then the maximum height from the centre of earth, to which the body can go, will be ( R : radius of earth)

To find the maximum height to which the body can go, we need to consider the gravitational potential energy and the kinetic energy of the body.

1. First, let's calculate the initial kinetic energy of the body. The kinetic energy (KE) is given by the formula KE = (1/2)mv^2, where m is the mass of the body and v is the velocity.

2. Since the body is projected vertically upward, the initial velocity (v) is equal to the escape velocity (ve). Therefore, the initial kinetic energy can be written as KE = (1/2)mve^2.

3. Next, let's calculate the potential energy at the maximum height. The potential energy (PE) is given by the formula PE = mgh, where m is the mass of the body, g is the acceleration due to gravity, and h is the height.

4. At the maximum height, the velocity of the body becomes zero. Therefore, all the initial kinetic energy is converted into potential energy. So, we can equate the initial kinetic energy to the potential energy at the maximum height: (1/2)mve^2 = mgh.

5. Now, let's solve for h, the maximum height. Dividing both sides of the equation by mg, we get (1/2)ve^2 = gh.

6. The acceleration due to gravity (g) can be calculated using the formula g = GM/R^2, where G is the gravitational constant, M is the mass of the Earth, and R is the radius of the Earth.

7. Substituting the value of g in the equation (1/2)ve^2 = gh, we get (1/2)ve^2 = (GM/R^2)h.

8. Rearranging the equation, we find h = (1/2)(ve^2)(R^2)/(GM).

9. Finally, substitute the given values of ve, n, and R into the equation to find the maximum height from the center of the Earth to which the body can go.