7.8. ESCAPE SPEED

If a stone is thrown by hand, we see it falls back to the earth. Of course using machines we can shoot an object with much greater speeds and with greater and greater initial speed, the object scales higher and higher heights.

This chunk introduces the concept of escape speed. It explains that if you throw an object, it will eventually fall back to Earth due to gravity. However, if you launch an object using a powerful machine, you can achieve much greater speeds, allowing the object to reach higher altitudes. The desire to understand if we can reach infinite height without falling back leads to the inquiry about escape speed.

Think of a toy rocket. When launched with a small force, it goes a little high and then falls back down. But if you use a powerful launcher, like a water rocket that builds up pressure, it can shoot very high into the sky. The height it can reach with enough force illustrates the idea of escape speed.

A natural query that arises in our mind is the following: ‘can we throw an object with such high initial speeds that it does not fall back to the earth?’ The principle of conservation of energy helps us to answer this question.

Here, we introduce the principle of conservation of energy, which tells us that energy cannot be created or destroyed but only transformed from one form to another. When considering escape speed, the total energy of an object at infinity (where it's no longer influenced by Earth's gravity) consists of its potential and kinetic energy. To escape Earth’s gravitational field, the energy at the point of escape must be equal to or greater than the energy required to overcome gravitational attraction.

Imagine you’re climbing to the top of a hill. To reach the top, you need to have enough energy (like food) to keep climbing up against gravity. Similarly, an object needs enough energy to escape Earth’s gravitational pull, making energy conservation crucial.

The energy of an object is the sum of potential and kinetic energy. As before W1 denotes that gravitational potential energy of the object at infinity. The total energy of the projectile at infinity then is 2 1 ( )2fmV E W∞ = +.

In this chunk, we focus on the mathematics behind energy. The energy of the object at infinity, when it can no longer feel Earth's gravity, is described as the sum of its kinetic energy (related to its speed at that point) and its gravitational potential energy (which is zero at that height). This sets the foundation for determining the minimum initial speed required to escape Earth's gravitational influence.

Think of throwing a ball. If it’s thrown lightly, it falls short, but if you throw it hard enough, it can go very high. The calculation of the energy is similar to measuring how hard you need to throw to ensure it doesn’t fall back.

If the object was thrown initially with a speed Vi from a point at a distance (h + RE) from the centre of the earth (RE = radius of the earth), its energy initially was 2 11( ) –2 ( )E i EGmME h R mV Wh R+ = ++.

This part defines the initial conditions of the launch, denoting the gravitational potential energy based on where and how an object is thrown (its distance from the center of Earth). This equilibrium of energy helps us set equations that will lead to determining specific escape speeds depending upon the height of launch.

It’s like placing a coin at the edge of a table versus dropping it from a height. The initial position matters greatly in how much energy it needs to escape the table (or Earth).

Thus, the minimum speed required for an object to reach infinity (i.e. escape from the earth) corresponds to ( )2 min1 2E i E GMmV h R= +.

This chunk defines the 'escape speed' mathematically, stating that the minimum speed an object must reach to escape Earth's gravity is dependent on the mass of Earth and its radius. When launched from the surface (h = 0), this simplifies to a standard equation for escape speed.

Think of riding a bicycle to escape a puddle. You need to reach a certain speed to go over it without slowing down and falling in. In a similar sense, the escape speed is the 'bicycle speed' for an object to overcome Earth's gravity.

Using the value of g and RE, numerically (Vi)min≈11.2 km/s. This is called the escape speed, sometimes loosely called the escape velocity.

This section finishes by providing the numeric value for Earth’s escape speed and contrasts it with that on the Moon. The moon has much lesser gravity, resulting in an escape speed that is significantly lower, displaying how gravity directly influences the escape speed for celestial bodies.

Consider how much effort it takes to jump on Earth versus how easy it is to jump on the Moon. If you think of escaping gravity as the jump speed needed, the Moon requires much less effort due to its weaker gravitational pull.

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