Using Newton's law of gravitation, derive a suitable formula for the escape velocity of an object at Earth's surface.

Newton's law of gravitation is;
F = GMm/(r²)
Where G is the Universal Gravitational constant, M is the mass of Earth, m is the mass of the object and r is the radius of Earth (no values are needed for this as we are simply deriving a formula, not working out a solution)
We can equate this force to the centripetal force experienced by an object at Earth's surface. This is because the centripetal force is what keeps an object in circular motion, acting towards the centre of the circle. It can be thought of as the force pulling us in toward the centre of the Earth, which we know is gravity so therefore is the same as the force given in Newtons law.
F = m(v²)/r (centripetal force)
Therefore;
GMm/(r²) = m(v²)/r
Dividing by m and multiplying by r
GM/r = (v²)
v = (GM/r)^1/2where v is the escape velocity