What is Energy - It's Introduction, Types, Formula, Examples, Conservation of Energy

Introduction:

Energy is the most fundamental concept, and it beautifully ties together work and power. The common dictionary definition, energy is the ability to do work. But energy is any quantity a number with the appropriate units i.e. Joules. But you can build intuition of energy as, Energy is the currency of the Universe. It’s the invisible "stuff" that makes everything happen, from the largest galaxy spinning to the smallest thought in your brain. Think of energy not as a thing, but as a currency, like money. It's an accounting system that measures the ability to do work, to cause change, or to make something happen. You can't hold a dollar bill and say this is value. A dollar bill represents value. In the same way, you can't hold a bottle of energy. But you can see what it can do.

Imagine energy as $100 in your pocket. This money (energy) has potential. You can transform it into different forms like pizza, a movie ticket, a book. You can transfer it to someone else. The total amount is conserved if you spend it, someone else gains it. You can't just create money out of nothing. Only fact we know about energy that it is conserved. Energy can be transformed from one form to another form. Other than that we know absolutely nothing else as to what energy is.

Suppose we have an isolated system. If it is not in equilibrium, its state may change. Energy conservation means that at the end of the change, the new state will have the same energy. Next we observe systems P and Q interacting. Before the interaction, P has 200 joules of energy. After interacting, A has 180 joules of energy, so it has lost 20 joules. Energy conservation says that if we measure the energy in system Q before and after the interaction, we will always find that system Q has gained 20 joules of energy. In general, system Q will always gain exactly however much system P loses, so the total amount is constant.

Energy is useful to us only if it has the ability to change its current form into another form. Energy can change from potential energy and into kinetic energy. For example, consider the static energy stored in the bonds of carbon and hydrogen in a gallon of gas. That bonding energy can be released and converted into useful kinetic energy, such as causing the relative motion of a car. Energy can also change from kinetic energy into potential energy and do useful work. Consider a ball rolling on a table; it has kinetic energy. If the ball collides with a spring, compresses it, and a latch catches it, the ball will lose its kinetic energy, and the spring will gain potential energy.

Energy primarily exists in two states:

1. Kinetic Energy:

Suppose that the force applied to move a crate is the only force acting on the crate in the direction of motion. What happens to the crate then? According to Newton’s second law, the crate will accelerate, and its velocity will increase. Doing work on an object increases its energy. What will we call the energy associated with the motion of the object. Yes, it's kinetic energy. It's a measure of how much change it can cause if it hits something or needs to be stopped.

Mathematical expression:

Imagine you are pushing a crate with wheels across the floor. Frictional forces are small enough to be ignored. We applied force F to move box through distance d. Because work involves the transfer of energy, the amount of kinetic energy gained by the crate should be equal to the amount of work done.

Kinetic Energy Gained = Work Done on the Object

Work done by a constant force is,
Work = Force × Distance
W = F * d

The force F we apply causes the box to accelerate. F = m * a.
W = (m * a) * d

Distance d is the box traveled while we were accelerating it in terms of its final velocity v.

When you accelerate an object from rest (0) to a final velocity (v), its average velocity is the sum of the start and end velocities divided by 2,
Average Velocity = (0 + v) / 2 = v/2

Distance = Average Velocity × Time
d = (v/2) * t

We need to relate Time (t) to Acceleration and Velocity.

Acceleration is the change in velocity over time,
a = (v - 0) / t
a = v / t
t = v / a

W = m * a * d
W = m * a * (v/2 *t)
W = m * a * (v/2 * v/a)
Acceleration cancel out,
W = m * (v/2 * v)
W = 1/2 mv²
KE = ½ m v²

The 1/2 comes from the fact that velocity isn't constant while the object is gaining energy. It starts at 0 and ends at v. 1/2mv² is the exact amount of work you must do to accelerate a mass m from rest to a speed v.


Kinetic energy depends on speed and mass:

• Mass:

Low mass means low kinetic energy, even at high speed. More mass means high kinetic energy.
e.g. A truck moving at 10 mph has more kinetic energy than a bicycle moving at 10 mph.

• Speed:

Faster speed means high kinetic energy, slower speed means low kinetic energy.
e.g. A car crashing at 60 mph have high kinetic energy than same car crash at 30 mph. This is why speeding is so incredibly dangerous.

e.g.
1. A bowling ball rolling slowly toward the pins might just nudge them. The same bowling ball thrown fiercely down the lane will smash the pins into pieces. The difference? Kinetic Energy. The faster ball has much more of it. Its motion has a bigger consequence.
2. You can gently tap a ping pong ball and it flies across the room. It's moving fast but has little consequence. If it hits you, you barely feel it. Low mass means low kinetic energy, even at high speed.
A baseball thrown at the same speed would sting your hand. A major league fastball could seriously injure you. High mass means high kinetic energy at the same speed. Now, imagine a baseball and a ping pong ball sitting still on a table. They have zero consequence. Their kinetic energy is zero.

Kinetic energy is energy in motion. It is an energy an object has because it is moving. Their motion does work on something else means transfer of energy.

e.g.

1. When you press the brakes, you are using friction to remove your car's kinetic energy. The faster you're going means more kinetic energy, the longer it takes to stop, and that energy is transformed into heat on your brake pads.

2. You swing the hammer to give it kinetic energy. When it hits the nail, it transfers that energy to do the work of driving the nail in.
3. The moving air (wind) has kinetic energy. When it hits the turbine blades, it transfers that energy to spin them, which is then converted into electrical energy.

Potential Energy:

Imagine a force that is continuously exerted on a body; for example, the gravitational interaction producing the body’s weight. To move upward against that downward pull requires the application of a counterforce that is provided by leg muscles or elevator and the doing of work. That gravitational force will still continue to act after the displacement. A body will still experiences a downward force while held up there at rest. When we loose the body, the gravitational force will drive it back down toward where it came from.

But what is happening while the body is held motionless, high in the gravitational field? When you do work to put an object into a special position or state, you are storing energy in it. This stored energy, energy by virtue of position or configuration in relation to a force, is known as potential energy (PE),

This is stored energy. It's energy an object has because of its position or state. It has the potential to be used later. Think of potential energy as an energy savings account. It's not energy being used right now, but energy that has been deposited and is waiting to be withdrawn for a specific purpose. Work done on a body against a conservative force such as gravity goes into changing the body’s PE, which may later go into changing its KE.

It's the energy an object has based on its position like being high up or its state like being stretched or having certain chemical bonds. It is potential because it's not active now, but it has the potential to be converted into kinetic energy and cause action in the future. It’s the paused moment, the drawn breath, the cocked hammer all the promise of action, stored and waiting.


Potential energy comes in a few common forms:

1. Gravitational Potential Energy:

Energy stored in body because of height. This is energy stored by lifting something up against gravity.
e.g. Imagine a rock sitting on the ground. It can't do much. Now, you lift it up and place it on the edge of a cliff. You did work against gravity to put it there. That work is now stored as gravitational potential energy in the rock. If the rock is pushed off the edge, gravity does work on it. Its stored potential energy is converted into kinetic energy, and the rock will fell off.

2. Elastic Potential Energy:

Energy stored by stretching or compressing something elastic.
e.g. Imagine an archer drawing a bowstring. The archer's muscles burn chemical energy to apply a force and pull the string back over a distance. This work isn't lost. Energy in the form of work is stored in the bent limbs of the bow as elastic potential energy. When archer releases the string. The stored potential energy is rapidly converted into kinetic energy, which is transferred to the arrow, launching it forward. The more you stretch or compress it, the more energy you deposit.

3. Chemical Potential Energy:

Energy stored in the bonds of molecules
e.g. Food, gasoline, a battery. They aren't moving, but their chemical structure holds energy that can be released to do work by burning or in a reaction. Battery just sit there. But they are packed with potential. The work was done by a battery charger to move electrons. This energy is stored in the chemical bonds. When you use the battery, a chemical reaction occurs, releasing the stored energy as electrical energy. Food is your body's chemical potential energy storage. You "withdraw" it to move, think, and live.

Mathematical expression:

1. For gravitational potential energy:

Imagine you lift a box straight up from the floor to a table. The box ends up at rest, so its kinetic energy hasn't changed. Where did the energy you spent go? It was stored as gravitational potential energy.

You are the external force doing the work. To lift the box at a constant velocity, you must perfectly oppose gravity. Therefore, the force you must apply is,
Force = Weight of the Box
= mass (m) × gravity (g)

Work done by you,
W = Force × Distance (d) × cos(θ)

You are lifting straight up,so the force and motion are in the same direction. cos(0°) = 1,
Work = F × d
Work = (m × g) × d

The distance you moved the box is its change in height. If the floor is height 0 and the table is height h, Distance = h

Work = (m × g) × h = m × g × h

All of that work was stored as energy. By definition, this stored energy is Gravitational Potential Energy (PE).

The potential energy gained by an object is equal to the work done by an external force to put it there.

PE = Work Done = m × g × h
= mgh

It depends on mass, gravity and height

1. Mass:

The heavier the object, the more work it takes to lift it, and the more potential energy it has. Lifting a bowling ball stores more energy than lifting a ping pong ball to the same height.

2. Gravity:

On a planet with stronger gravity, the same object would be heavier. It would take more work to lift it, so it would have more potential energy at the same height. On the Moon, where gravity is weaker, objects have less potential energy at the same height. On the Earth, where gravity is stronger, objects have high potential energy at the same height.

3. Height:

The higher you lift something, the more work you do, and the more potential energy it stores. A brick held at your waist has less potential energy than a brick held over your head. Here the path doesn't matter. Whether you lift the box straight up or take it on a complicated ramp journey to the same height, the total work you do against gravity is always mgh.

2. For elastic potential energy:

You cannot use mgh for elastic potential energy. They are fundamentally different types of storage, governed by different forces.

Gravity for small height changes provides a Constant Force. The force of gravity on an object weight = mg is essentially constant near Earth's surface. It doesn't matter if the object is at your knees or at your head; gravity pulls down with the same force. This simplicity is why the workdone gives us the PEgrav = mgh

While a Spring provides a Changing Force. The force required to stretch or compress a spring is not constant. The more you stretch it, the harder it pulls back, by Hooke's Law: F = -k * x
At the start, it's easy to compress the spring means low force. At the end, it's very difficult to compress it further high force.

Because the force changes, we can't just do Force × Distance. We have to find the average force you applied over the distance.

The spring's force,
Fspring = k * x,
where k is the stiffness and
x is the distance from its relaxed position.

To compress it,you must apply an equal and opposite force,
Fcomp= k * x

The Force is Not Constant. The force you need to apply starts at 0, when x=0 and ends at k * x, at the final compression.

Find the Average Force Because the force increases linearly from 0 to k*x,
F_avg = (initial + final ) / 2 = (0 + k*x) / 2 = (½)k*x

Work done to compress the spring a distance x,
Work = Average Force × Distance
= (½k*x) * x = ½ k x²

The potential energy stored in a spring is equal to the work you did to stretch/compress it,
PEelastic = ½ k x²

The ½ comes directly from the fact that you weren't fighting the full force k*x for the entire distance. You started with a very small force and ended with the maximum force.

Imagine lifting a Box with a Weak Spring against gravity,but gravity gets stronger the higher you lift. The first meter is easy, but the second meter is very hard. The average force you have to apply is only half of the final force at the top. Therefore, the work you do is,
½ × Final Force × Height.

So, while both mgh and ½kx² represent stored work, they come from different physical situations. mgh is for storing energy by moving against a constant force i.e. gravity. While ½kx² is for storing energy by moving against a force that increases linearly with distance i.e. a spring's resistance. You cannot use mgh for a spring because the force isn't constant.

Conservation of Energy:

It is not just a theory but a fundamental law of the universe. Only thing you have to remember that energy doesn't disappear, it just changes form. Think of the total amount of energy in the entire universe as a fixed number of coins. This number cannot change. Not a single coin can be created or destroyed. What can change is what form those coins take and who holds them. Suppose a system having energy transfer between two bodies A and B by any means of force. If body A loses energy then obviously body B gains energy.

Mathematical expression for the conservation of energy is,
KE₁ + PE₁ = KE₂ + PE₂

More generally,
ΣE_initial = ΣE_final

Imagine a system like a roller coaster car, has a fixed currency of energy. This source contains two types currency: Kinetic Energy (KE) and Potential Energy (PE).

You can trade between these currencies at any time, but the total value of the fund must always stay the same. Mathematical way of stating this rule is,

KE₁ + PE₁ = KE₂ + PE₂

e.g.
At the top of the first hill, the coaster is barely moving. It has almost no kinetic energy. But it is very high up. It has a huge amount of gravitational potential energy by virtue of height.

Total Energy = Large Amount of Potential Energy + No Kinetic Energy

At halfway down the hill, the coaster is picking up speed. It has lost some height, so it has less potential energy. But that energy didn't disappear, It was transformed into kinetic energy.

Total Energy = half Potential Energy + half Kinetic Energy

Buy the sum of these two values is exactly the same as the potential energy it had at the top.

At the bottom of the hill, the coaster is at its lowest point and moving at its maximum speed. It has almost no potential energy left. But it has a massive amount of kinetic energy.

Total Energy = No Potential Energy + Maximum Kinetic Energy

This kinetic energy is exactly equal to the potential energy it started with.

If you could remove all friction and air resistance, the coaster would go up and down forever, with energy sloshing back and forth between potential and kinetic, never losing a single joule.

In real world this doesn't happen in real world. Because there is difference between the idealized world of textbook physics and the messy reality of our universe. Energy often dissipates as heat due to friction or other non-conservative forces. But the conservation law still holds universally if we account for all forms of energy, including heat. Energy isn't truly leaking away in the form of heat it's just transforming into a less useful form. The roller coaster example with friction
would work well here by showing how
mechanical energy decreases but total
energy including thermal remains constant.

The first time the coaster goes down the hill, KE at bottom = PE at top. But because of air resistance and friction in the wheels which generate heat, when it goes up the next hill, it won't quite make it back to its original height.
KE + PE at the bottom is less than the PE at the top. This is not a violation.

The missing mechanical energy is exactly accounted for by the thermal energy generated by friction i.e. ΔE_{thermal}

Total Energy is still conserved: PE_initial = KE_final + PE_final + Heat and Sound

The energy is still there, but it's now dissipated as heat into the environment. It's no longer useful for moving the car, but the cosmic accountant still counts it.

The simple formula KE₁ + PE₁ = KE₂ + PE₂ is for a closed system with no non-conservative forces like friction. It's a ideal system.

The real, more general mathematical expression that accounts for the "leak" is:

KE₁ + PE₁ + W{other} = KE₂ + PE₂ + ΔE_{thermal}

The mechanical energy we start with,plus any extra work we put in like from an engine, equals the mechanical energy we end up with, plus all the energy that was leaked and turned into heat.

Imagine car is moving. KE₁ is high. PE₁ is unchanged. You apply break. Car is stopped. KE₂ = 0.

The friction between the brake pads and the wheels did negative work i.e. W{friction} on the car. This negative work didn't destroy the car's kinetic energy. It transformed it.

KE₁ + W_{friction} = 0
KE₁ = -W_{friction}

This tells us the amount of kinetic energy we started with is exactly equal to the amount of work friction did. And what did that work produce? Heat. The brakes get very hot because of thermal energy. The car's initial kinetic energy was transformed into an equal amount of thermal energy in the brakes and the air.

In any real-world process, we never get all the useful energy back. Some of it always seems to "leak" away as heat. This doesn't break the Law of Conservation of Energy. The law doesn't promise that energy will stay in a useful form, only that it is never destroyed. We don't have an energy crisis; we have an energy quality crisis. The universe's total energy is constant, but it is constantly becoming more dispersed and less useful for doing work.