ESCAPING THE BONDS OF EARTH: ESCAPE VELOCITY
Isaac Newton brilliantly equated gravity to the mass of our planet. Newton's Law of Universal Gravitation states that the force experienced by an object of mass, m, is:
Where r = distance from the Earth's center, M is the mass of the Earth, and G, a constant whose value Newton was not able to determine at the time (it was experimentally determined in 1798 by Henry Cavendish, 71 years after Newton died).
Newtonian Physics views gravitational forces as arising from a force field that surrounds any mass, m. The gravitational force field of the Earth occupies the space around the planet.
The gravitational field, represented in the diagram at left as gray shading, is strongest near the surface and diminishes as we move away from the planet. The equation above suggests that the Force weakens with the square of the increasing distance:
This feature is due both to the shape of the Earth and to the nature of force field (the field must exit perpendicular to the surface of a mass). The gravitational field of a sphere follows what is called the "Inverse Square Law": if you double the distance from the center of the mass, the force reduces by a factor of 1/4. If you triple the distance, the force is 1/9 of its original value. Quadruple the distance and the force is 1/16 of its original value. And so on. A way to visualize this is by using "field lines": the field lines indicate the direction of the force. Notice two features: 1) the lines are perpendicular to the Earth's surface (making them radial); 2) the lines are further apart as you move away from the Earth; this means the field gets weaker as you move away (this is a result of the object's spherical shape…what would happen if the Earth were flat?)
If a rocket motor exerts a greater force away from the center of the Earth than the force with which the Earth holds the rocket (essentially its weight), it will accelerate upwards. To escape from the Earth, this rocket must expend energy. We consider two types of energy:
1. Kinetic energy (KE) is the energy of motion: m = mass, and v = the velocity, or speed, of the object
2. Potential energy (PE) is the energy of position (or configuration); the higher you lift a book off the floor, the greater the energy it has relative to the floor (and the more it will hurt if it lands on your foot).
(Gravitational) Potential Energy (PEgrav) is a special form of Potential Energy. When moving far above the Earth's surface it can be estimated using this equation:
Where m = mass of the rocket, or other object; M = mass of the Earth, r = distance from the center of the Earth; and G = universal gravitational constant.
The Earth's gravitational field is a "potential well". If we were to visualize the Earth's gravity as a potential well, it would look a bit like a shallow funnel:
The Earth is at the throat of the funnel. An object on the rim will slowly 'fall' into the funnel, picking up speed as it moved. To climb out, we need to climb up past the rim. Relativistic physics (Einstein's Theory of General Relativity) embraces the view that space is 'warped', like the funnel above, in the presence of matter (mass).
We must use energy if we want to climb out of the well. One way to climb out of the well is to exchange speed for height. It is like throwing a ball straight up into the air: the harder it is thrown, the higher it goes. How fast do we have to throw the "ball" so that it never falls back down?
To calculate this we use the Conservation of Energy Law. The law states that energy cannot be created or destroyed - it only changes form. The energy of a system remains constant:
Initial Rocket Energy + Energy Lost thru heat and air friction = Final Rocket Energy
Our “system” is the Earth and the rocket. We will ignore energy losses, such as heating and air friction, to simplify the discussion. So we can re-write it this way:
Initial Rocket Energy = Final Rocket Energy
The initial rocket energy is made up of its Kinetic Energy (speed) and its Gravitational Potential Energy (which is the well we must climb out of).
(KE + PEgrav) of rocket on earth = (KE + PEgrav) of rocket far, far into space
Everything on the right side of the equation is zero: the gravitational potential energy (PEgrav) is zero since we want to escape Earth's gravity. Likewise, by that time, we will have used up all of our speed in getting there, so the final kinetic energy (KE) is also zero (remember: we are looking for the minimum escape speed…if we had excess energy it just means we’ll be moving fast even after we’re beyond Earth’s influence). Therefore,
KE + PE of rocket on Earth = 0 + 0
Substituting in the numbers: the gravitational constant, G = 6.67 x 10-11 N·m2/kg2, for Earth's mass, M = 5.98 x 1024 kg, and r = radius of Earth = 6.38 x 106 meters. This results in the escape velocity:
vescape = 11,200 meters/second
If a rocket leaves the surface of the Earth with this initial velocity, it will escape Earth's gravity just as its velocity slows to zero. Again, if we start with a greater velocity, then we won’t slow to a stop.
Teacher-Directed Activities and Demonstrations:
1. Inverse Square Law Demo (a visual to show how gravity diminishes as you move away from the Earth)
2. Gedanken Experiment (a thought experiment, inspired by Einstein, of gravitational effects at the center of the Earth)
3. G-forces on Blast-Off!: (student pre-requisite: constant acceleration equations; students read part or all of the online version of Jules Verne’s “From the Earth to the Moon")
Students!: you can read this story on your own at: http://jv.gilead.org.il/pg/moon/