Derivation Of Physics Formulas

List of Derivations of Physics Formulas

Physics is a subject that is fundamentally based on mathematical equations and their derivations. These derivations help us understand the underlying principles and concepts of physics. Here are some of the key physics formulas and their derivations:

1. Newton’s Second Law $(F=ma)$: This law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. The derivation of this formula is straightforward as it is a definition.

2. Gravitational Force $(F=G(m_1m_2)/r^2)$: This formula is derived from Newton’s law of universal gravitation. Here, $F$ is the force of attraction between the two bodies, $m_1$ and $m_2$ are the masses of the two bodies, $r$ is the distance between the centers of the two bodies, and $G$ is the gravitational constant.

3. Kinetic Energy $(KE=\frac{1}{2}mv^2)$: This formula is derived from the work-energy theorem. The work done on an object is equal to the change in its kinetic energy. Here, m is the mass of the object and v is its velocity.

4. Potential Energy $(PE=mgh)$: This formula is derived from the work done against gravity to lift an object to a certain height. Here, m is the mass of the object, g is the acceleration due to gravity, and h is the height.

5. Ohm’s Law $(V=IR)$: This law states that the voltage across a resistor is directly proportional to the current through it. The constant of proportionality is the resistance. This formula is derived from the definition of resistance.

6. Einstein’s Energy-Mass Equivalence $(E=mc^2)$: This formula is derived from Einstein’s theory of relativity. It states that the energy of an object is equal to its mass times the speed of light squared. Here, E is the energy, m is the mass, and c is the speed of light.

7. Snell’s Law $(n_1sinθ_1 = n_2sinθ_2)$: This law describes the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media. Here, $n_1$ and $n_2$ are the refractive indices of the two media, and $θ_1$ and $θ_2$ are the angles of incidence and refraction, respectively.

These are just a few examples of the many physics formulas and their derivations. Each of these derivations is based on fundamental principles and laws of physics, and understanding them can provide a deeper understanding of the subject.

Advantages of Derivation Of Physics Formulas

Derivation of physics formulas is a crucial aspect of physics that offers numerous advantages. It involves the process of obtaining a formula by starting from basic principles and laws of physics. Here are some of the benefits of deriving physics formulas:

1. Understanding the Basics: Derivation of physics formulas helps in understanding the basic principles and laws of physics. It allows students to understand how a particular formula is derived and the principles behind it. This understanding is essential for applying the formula correctly in different situations.

2. Problem Solving: Deriving formulas can help in problem-solving. Often, in physics, we encounter problems that cannot be solved directly using standard formulas. In such cases, understanding the derivation of formulas can help in modifying or adapting them to solve the problem.

3. Critical Thinking: The process of deriving formulas involves logical reasoning and critical thinking. It helps in developing these skills, which are not only important in physics but also in other areas of life.

4. Application in Research: In research, new situations and problems often arise that require the development of new formulas or modification of existing ones. Understanding the derivation of formulas can be very helpful in such situations.

5. Avoiding Memorization: If you understand how a formula is derived, you don’t need to memorize it. You can always derive it when needed. This not only reduces the burden of memorization but also ensures that you understand the formula and its application better.

6. Building a Strong Foundation: Deriving formulas helps in building a strong foundation in physics. It helps in understanding the interconnections between different concepts and principles, which is essential for a deep understanding of the subject.

7. Enhancing Mathematical Skills: The derivation of physics formulas often involves mathematical operations and techniques. Therefore, deriving formulas can also help in improving mathematical skills.

In conclusion, the derivation of physics formulas is an essential part of learning physics. It helps in understanding the subject better, improves problem-solving and critical thinking skills, and can be very useful in research. It also reduces the need for memorization and helps in building a strong foundation in the subject.

Some Important Derivations:

Deriving physics formulas involves using fundamental principles and mathematical reasoning to arrive at equations that describe physical phenomena. Below are a few examples of common physics formulas and their derivations:

1. Kinematic Equations for Uniformly Accelerated Motion

One of the most fundamental sets of equations in physics describes the motion of an object under constant acceleration. The three main kinematic equations are:

1. $ v = u + at $
2. $ s = ut + \frac{1}{2}at^2 $
3. $ v^2 = u^2 + 2as $

Where:

• $ u $ = initial velocity
• $ v $ = final velocity
• $ a $ = acceleration
• $ t $ = time
• $ s $ = displacement

Derivation of the First Equation: $ v = u + at $

1. Start with the definition of acceleration: $$ a = \frac{v - u}{t} $$ Rearranging gives: $$ v = u + at $$

Derivation of the Second Equation: $ s = ut + \frac{1}{2}at^2 $

1. Use the average velocity: The average velocity $ v_{avg} $ during the time $ t $ is given by: $$ v_{avg} = \frac{u + v}{2} $$
2. Substituting $ v $ from the first equation: $$ v_{avg} = \frac{u + (u + at)}{2} = \frac{2u + at}{2} = u + \frac{1}{2}at $$
3. Displacement: $$ s = v_{avg} \cdot t = \left(u + \frac{1}{2}at\right)t = ut + \frac{1}{2}at^2 $$

Derivation of the Third Equation: $ v^2 = u^2 + 2as $

1. Start with the first equation: $$ v = u + at $$
2. Square both sides: $$ v^2 = (u + at)^2 = u^2 + 2uat + a^2t^2 $$
3. Substitute $ t $ from the second equation: From $ s = ut + \frac{1}{2}at^2 $, we can express $ at $ in terms of $ s $: $$ at = \frac{2(s - ut)}{t} $$ However, a more straightforward approach is to eliminate $ t $ directly: From $ s = ut + \frac{1}{2}at^2 $, we can rearrange to find $ t $ in terms of $ s $: $$ s = ut + \frac{1}{2}at^2 \implies 2s = 2ut + at^2 $$ Rearranging gives: $$ at^2 + 2ut - 2s = 0 $$ Solving this quadratic equation for $ t $ and substituting back into the squared equation leads to the final result: $$ v^2 = u^2 + 2as $$

2. Newton’s Second Law: $ F = ma $

Newton’s second law states that the force acting on an object is equal to the mass of that object multiplied by its acceleration.

Derivation:

1. Start with the definition of acceleration: $$ a = \frac{F_{net}}{m} $$ Rearranging gives: $$ F_{net} = ma $$

3. Ohm’s Law: $ V = IR $

Ohm’s Law relates voltage (V), current (I), and resistance (R) in an electrical circuit.

Derivation:

1. Start with the definition of resistance: $$ R = \frac{V}{I} $$ Rearranging gives: $$ V = IR $$

These derivations illustrate how fundamental principles in physics can be used to derive important formulas. Each derivation relies on basic definitions and relationships, showcasing the interconnectedness of physical concepts. Understanding these derivations helps in grasping the underlying principles of physics and applying them to solve problems.