# Gravitational Law - Very Difficult

• mitch_nufc
In summary, the conversation discusses an object moving vertically upwards from the surface of a planetary body under the action of Newton's Gravitational Law without resistance. The equation z'' = -gR^2 / (z + R)^2 is used to find the relation between v = z' and z and obtain a numerical estimate for the escape speed on the surface of the Earth. The conversation also mentions the use of the chain rule and the conservation of energy in solving the problem.

#### mitch_nufc

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 can't 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

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

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

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 don't 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'

## What is the Gravitational Law?

The Gravitational Law, also known as Newton's Law of Universal Gravitation, is a scientific law that describes the force of gravity between two objects. It states that every object in the universe is attracted to every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

## Why is the Gravitational Law considered difficult?

The Gravitational Law is considered difficult because it involves complex mathematical equations and concepts, such as calculus and the concept of a gravitational field. It also requires a deep understanding of physics and the laws of motion.

## How is the Gravitational Law related to other scientific laws?

The Gravitational Law is related to other scientific laws, such as the Laws of Motion and the Law of Conservation of Energy. These laws work together to explain the behavior of objects in the universe, including the effects of gravity.

## What are some real-world applications of the Gravitational Law?

The Gravitational Law has many real-world applications, including predicting the motion of planets and other celestial bodies, calculating the orbits of satellites, and understanding the behavior of objects in free fall. It is also essential in fields such as astronomy, physics, and engineering.

## Are there any exceptions to the Gravitational Law?

While the Gravitational Law is incredibly accurate in explaining the behavior of objects in most situations, it does have limitations. It does not apply to objects at the atomic or subatomic level, and it does not take into account the effects of relativistic speeds or extreme gravitational fields, such as those near black holes.