# How Do You Solve For Conservation Of Energy?

The law of conservation of energy is one of the fundamental concepts in physics. It states that the total energy of an isolated system remains constant and is said to be conserved over time. This means that energy can neither be created nor destroyed – it can only be transferred or changed from one form to another.

Understanding conservation of energy is crucial for analyzing physical systems and solving problems in mechanics, thermodynamics, electromagnetism, optics and other areas of physics. The concept allows tracking the transfers and transformations of energy within a system during processes and interactions. By applying conservation of energy, unknown variables and quantities can be determined.

## Understanding the Problem Statement

The first step in solving any physics problem that involves conservation of energy is to clearly identify the system you are analyzing. The system refers to the specific object or objects that are involved in the interactions described in the problem.

For example, if the problem states that a ball is dropped from a tower, the system would be the ball. The surroundings refer to everything outside of the system. In our example, the surroundings would be the tower, the air, and the ground.

It’s also crucial to identify the different types of energy involved in the problem. This depends on the system you defined. Common types of energy include gravitational potential energy, kinetic energy, thermal energy, elastic potential energy, electric potential energy, etc.

Understanding the system and types of energy is key because according to the law of conservation of energy, the total energy of an isolated system remains constant. Energy cannot be created or destroyed, only transferred between different objects and converted from one form to another.

## Drawing a Diagram

When solving any physics problem, it’s always helpful to draw a pictorial representation of the situation. This allows you to visualize the problem and identify the knowns and unknowns. For conservation of energy problems, your diagram should include:

- An object or system of interest
- The initial energy forms and values (e.g. gravitational potential energy)
- The final energy forms and values (e.g. kinetic energy)
- Arrows indicating energy transfers or transformations

The diagram brings the verbal description to life and serves as a useful reference as you proceed through the solution. Don’t forget to label all energies with the appropriate variables. A well-drawn diagram is a vital first step in successfully solving conservation of energy problems.

## Writing Out the Given Information

The first step in solving any physics problem that involves conservation of energy is to carefully write out what is known from the problem statement. This involves identifying the system you are analyzing, listing the known quantities, and determining what needs to be solved for.

For example, if the problem states that a ball is dropped from a height of 5 meters above the ground, the known information is:

- The system is a ball.
- The initial height of the ball is 5 meters.
- The final height of the ball on the ground is 0 meters.

Carefully writing out the known quantities focuses your attention on the givens and helps identify what needs to be determined in the problem – in this case, you need to solve for the final velocity of the ball. Having the known information neatly organized will make applying the conservation of energy equation much easier.

## Applying Conservation of Energy

The key principle for solving conservation of energy problems is to equate the total initial energy in a system to the total final energy. This is known as the conservation of energy equation:

Initial Total Energy = Final Total Energy

This fundamental equation states that the total amount of energy in a closed system remains constant. While energy can change forms within the system (such as potential energy converting to kinetic energy), the total quantity of energy is the same at the beginning and end.

To apply conservation of energy, you first need to identify the system you are analyzing. The system refers to the specific object or objects you are tracking the energy for. Once the system is defined, sum up all the types of energy it has initially. This is the total initial energy.

Next, determine the different types of energy the system has at the end, after some process has occurred. Sum these up to find the total final energy.

Finally, set the initial total energy equal to the final total energy. Now you have an equation with one unknown that can be solved to find the value of the unknown energy.

## Substituting the Knowns

Once the diagram is drawn and the known information is written out, the next step is to plug in the numerical values for each energy term in the conservation of energy equation. For example, if the problem stated that a ball was dropped from a height of 5 meters with an initial gravitational potential energy of 200 Joules, then you would substitute:

Initial gravitational potential energy = 200 J

Final gravitational potential energy = 0 J (because the final height is 0 m)

Kinetic energy = unknown

By plugging in the known numerical values into the conservation of energy equation, you have:

200 J + 0 J = Kinetic energy

This allows you to solve for the unknown kinetic energy term algebraically in the next step.

## Solving for the Unknown

After setting up the conservation of energy equation and substituting all the known values, the next step is to isolate and solve for the unknown variable. To do this:

- Use algebra to isolate the unknown variable on one side of the equation and all other terms on the other side.
- For example, if your equation is mgh1 + 1/2mv12 = 1/2mv22, move all terms without v2 to the left side and v2 terms to the right side.
- This gives mgh1 + 1/2mv12 – 1/2mv22 = 0.
- Factor out the unknown variable v2.
- Divide both sides by the coefficient of v2.
- This gives v2 = 2(mgh1 + 1/2mv12) / m.
- You now have an equation with only the unknown variable on one side that can be solved.

Using algebraic manipulation to isolate the unknown variable is a key step in solving any conservation of energy problem. Mastering this will allow you to find the missing information needed to fully solve the problem.

## Checking Units

When solving any physics problem, it is critical to check that the units are consistent on both sides of the equation before proceeding. The most common way to check is by going through each term and ensuring the units match.

For example, if your equation contains a velocity term with units of m/s, then every other term must also contain units that are compatible with m/s when combined. The goal is to end up with the same units on both sides of the equals sign.

If the units do not match, then a mistake has likely been made, and the problem needs to be re-examined before continuing. Having the correct units is a good quick check that the logic and math are sound. It’s an easy step that can prevent wasted effort from trying to solve an equation that is fundamentally flawed due to inconsistent units.

## Interpreting the Result

Once you have solved for the unknown variable in the conservation of energy equation, it is important to interpret what the numerical solution physically represents. The value obtained is not just an abstract number, but rather describes a real parameter in the physical system. For example, if you solved for the final velocity v of an object, the velocity value tells you how fast the object is moving after some process occurred. Or if you solved for the height h, this reveals how high the object reached. The numerical solution always connects back to the actual observable properties of the system.

Therefore, take a moment to reflect on what the solved variable embodies in the real world scenario depicted in the problem. Ensure the magnitude and units make sense based on your intuition and expectations. This builds a deeper conceptual understanding beyond just crunching through the mathematical derivation. It also provides an opportunity to catch any potential mistakes before moving forward with the solved value.

Connecting the quantitative result to the qualitative physical meaning is an essential part of applying conservation of energy principles. The goal is not just to solve for numbers, but to gain insight into how energy transfers and transforms within a system according to the governing laws of physics.

## Example Problems

Let’s go through a couple examples to see how to apply conservation of energy to solve for unknown values.

**Example 1:**

A block with mass 2 kg is released from rest at a height of 5 m above the ground. Ignoring air resistance, determine its speed just before hitting the ground using conservation of energy.

*Solution:*

First, let’s draw the diagram:

The initial energy Ei is just potential energy:

Ei = mgh

= (2 kg)(9.8 m/s2)(5 m)

= 98 J

The final energy Ef is just kinetic energy:

Ef = 1/2mv2

By conservation of energy, Ei = Ef

98 J = 1/2(2 kg)v2

v = √(196/2) = 7 m/s

Therefore, the final speed is 7 m/s.

**Example 2:**

A car starts from rest and accelerates at 2 m/s2 for 5 seconds. Determine the final speed using conservation of energy.

*Solution:*

The initial energy Ei is 0 since the car starts at rest.

The final energy Ef is kinetic energy:

Ef = 1/2mv2

We know:

m = mass of car

v = at

= (2 m/s2)(5 s)

= 10 m/s

Substituting this into the kinetic energy equation:

Ef = 1/2(m)(10 m/s)2

= 50mv J

By conservation of energy, Ei = Ef

0 = 50mv J

Therefore, the final speed is 10 m/s.