What is Energy?

This topic focuses on the generation of Energy that can be boiled dow to Newton’s Theories. Energy is essentially the ability to do work. It does not disappear, just changes to another form. It’s not a physical thing and we also measure Energy in many forms.

Types of Energy:

  • Chemical – Primary
  • Nuclear – Primary
  • Heat (s0lar) – Primary
  • Renewables – Primary
  • Gravitational – Secondary
  • Electrical – Secondary
  • Kinetic – rotational – Secondary
  • Kinetic – linear – Secondary

Primary Energy Sources: Harnessed directly from nature

Secondary Energy Sources: Resources that have previously been converted or stored

Energy can be converted.

For the sake of simplicity, think of the human body. Food contains chemical energy, we eat it and the body converts it to kinetic energy so we can move. Plus additional heat energy, therefore, we are warm. The case is similar with gunpowder.

How Energy and Work are measured

The standard unit to measure work and Energy is the joule (J).

As mentioned above, we measure Energy in many forms. When we transfer Energy in a form of Force we talk about Work. An example when moving an object.

1 joule is the energy transferred when a force of 1 Newton moves an object 1 metre.

Other form of Energy is the Calorie. 1 Calorie = 4.184 joules.
1 Calorie is the energy required to heat 1 g water by 1°C.

Work Done

Work (W) is done if a force (F) acts on an object over a distance (x):

W = F × Δx

If we plot the Force over Distance graph, the energy is the area under the curve. In this example, for simplicity, the force is constant over the distance, thus E = F × x.

Energy Graph: Distance vs Force. Energy is equal to the area under the curve.

We also get the area under the curve using integration. It is especially used when the area is hard to calculate by common means. So, the above equation can be also written as:

Equation: Energy is the integral of force with respect to time

What integration does, is that it cuts up the area into smaller chunks and adds them up. The force is integrated with respect to distance.

Gravitational Potential Energy

Objects that are high above ground have gravitational potential energy. It is defined as:


E = m×g×h

where:

  • m – mass of object in kg
  • g – local gravitational field of Earth
  • h – height of the object above ground in meters

So, adding up what we know:

Equation: Energy relationship to gravitational field, mass and height

There is one problem with E = mgh though. It only works if we have a small h compared to the Earth’s radius. This is because in great heights g changes.

Newton’s Universal Law of Gravitation

Newton’s Universal Law of Gravitation describes the Force that two objects have on each other. This is a gravitational force:

Equation: Newton's Universal Law of Gravitation

Where:

  • F – force
  • G – gravitational constant
  • M – mass of object 1
  • m – mass of object 2
  • r – distance between the centres of the objects

Let’s translate this to the case of the object above the Earth. For this the mass of Earth will be M and the mass of the object m.

We are trying to find the potential Energy that the object would lose if it fell on the surface of the Earth. This distance is between the surface and the object.

Above we described that Energy is the integral of Force over a distance. Let’s integrate Newton’s Universal Law of Gravitation for the distance between the surface of Earth and the object.

In the equation r is the distance between the centres of the objects. Since we are integrating from the surface of the Earth, the lower limit will be the Earth’s radius Re.

Calculation: Using Newton's Universal Law of Gravitation, we derive the Potential Energy from force by integration

Note, if you are foggy with integration, first we took the constants outside of the integral. If we have dr in the integral, we are integrating with respect to r, the rest is constant. Next we use the Power Rule:

Integration Power Rule

Since we have a definite integral we can ignore the +C constant. Lastly, we first substitute the upper limit and we subtract from this the substitution with the lower limit.

So, to calculate the Energy the object would lose if it fell from a height h onto the surface of the Earth we could use the above derived equation:

Equation: Energy the object would lose if it fell from a height h onto the surface of the Earth

Where:

  • E – energy
  • M – mass of Earth
  • m – mass of object
  • h – height of object
  • Re – radius of Earth:

Escape Velocity

The Escape Velocity is the minimum velocity that an object needs on the surface of the Earth to escape the planet’s gravitational pull. In other words, to be able to fly out to space.

This can easily be calculated the same way as above. We use h = ∞

Calculation: Potential Energy between Earth's surface and infinity

So, this is the potential energy. At the height of infinity the potential Energy = 0 as we are very far from the Earth with no gravitational pull. This means the initial kinetic Energy will be equal to the potential Energy.

Ve is the escape velocity we are looking for.

Calculation: Escape Velocity

We have a known equation for g, thus substituting we get the escape velocity:

Calculation: Simplifying the Escape Velocity Equation

Derivation of the Kinetic Energy Formula

We know Force is mass x acceleration.

We also know that velocity is acceleration x time.

Note, velocity is essentially speed that has direction. Speed is a scalar quantity, whereas velocity is a vector quantity.

From these relationship we can define the velocity like so:

Equation: Deriving relationship between Velocity, Force, Mass and Time

We also know that distance is the integral of the velocity over time. Why? If we investigate the plot below, what we see is an object (car) changes velocity over time.

The distance this object takes all together is equal to the area under the curve. This is a fairly more complicated version of the speed = distance/time formula. Since the area is not simple to calculate, we use integration, which essentially means summing up the different parts under the curve.

Graph: Time vs Velocity - Distance is equal to the area under the curve

The distance can be expressed in the following way. We substitute v from above and integrate from 0 to T using the Power Rule as above.

Calculation: deriving a formula for distance from velocity using integration with respect to time

We know that Energy is essentially Force x distance. (note we used x for distance above, we use d here. Same thing).

Substitute d and solve the equation.

Calculation: Deriving the Kinetic Energy equation

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