# Gravitational Law - Very Difficult

An object is fired vertically upwards from the surface of a planetary body; it moves under the action of Newton’s Gravitational Law, without
resistance, so the equation is z'' = -gR^2 / (z + R)^2 . Find the relation between v = z' and z and use this model, and the relation that you have
found, to obtain a numerical estimate for the escape speed on the surface of the Earth.

Its the wording of the question I don't get? I assume I integrate z'' twice to find the velocity and distance respectively, but i already have a z on the RHS so i cant integrate, but the escape speed to escape the surface of the earth? is this like when g (9.81) begins to decrease as it moves from the planetry body? I havn't got a clue at all. Any help appreciated

Cyosis
Homework Helper
The escape velocity is the velocity at which the kinetic energy is equal to the gravitational potential energy of the planet.

Using the chain rule you can write $$\frac{d^2 z}{dt^2}=\frac{d v}{dt}=\frac{dv}{dz}\frac{dz}{dt}=v\frac{dv}{dz}$$ After rearranging your equation you can now integrate.

How did you get the equation for z''?

it was given in the question, the equation for z'' that is

Cyosis
Homework Helper
Odd that the mass of the planet isn't a part of the equation.

Does it help to know I'm doing a Maths degree and not a Physics one? We were told none of our questions would require knownledge of classical mechanics/physics, but just differential equations etc

Cyosis
Homework Helper
Regardless of what they said this is somewhat of a physics problem, but it matters not. Just use the equation you were given.

Hi there,

Your first equation: $$\frac{d^2z}{dt^2} = ...$$ is not Newton's gravitational law. It is the result of the Newton's gravitational law, considering a planetary body, like the Earth, the Sun, or the Moon. Newton's real gravitational law is define as the force that attracts two body of masses, and is express with : $$F_g = G\frac{m_1 \cdot m_2}{r^2}$$ where $$G$$ is Universal gravitational constant, $$m_i$$ are the mass of each body, and $$r$$ is the distance between the two body's center of mass.

Hope this helps you understand it more.

yeah i remember all this from A-level physics, but i dont know any masses or distances so i assume theyre kept as constants in the ODE

There is no mention to the mass of the planet because it's inside g, the formula is correct (physically speaking). If this was merely a maths problem, you'd have to solve the ODE with the initial conditions z(0)=0, z'(0)=v and find out for which v z(infty)->infty. But this is too much work for a physicist... We know that there is a conserved quantity, energy:

$$E=z'^2 - {gR^2\over z+R}$$

and that for E=0, the particle will escape, so... we set z=0 in that equation and solve for z'