Physics Formulas Derived

Introduction to Derived Physics Formulas

Physics is built upon fundamental principles expressed through mathematical equations. Many complex formulas are derived from basic physical laws, allowing us to solve real-world problems efficiently. In this article, we will explore key derived physics formulas and their derivations.

1. Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. These equations are derived from the basic definitions of velocity and acceleration.

Deriving the First Equation of Motion

The acceleration is defined as:

\[ a = \frac{v - u}{t} \]

Rearranging for final velocity:

\[ v = u + at \]

Where:

v = Final velocity
u = Initial velocity
a = Acceleration
t = Time

Deriving the Second Equation of Motion

Using the formula for displacement:

\[ s = ut + \frac{1}{2}at^2 \]

This is derived by integrating the velocity equation over time.

Deriving the Third Equation of Motion

We use:

\[ v^2 = u^2 + 2as \]

This is derived by eliminating time \( t \) from the first two equations.

2. Work-Energy Theorem

The work done on an object is equal to the change in its kinetic energy:

\[ W = \Delta KE = \frac{1}{2} m v^2 - \frac{1}{2} m u^2 \]

This is derived using Newton’s Second Law and kinematic equations.

3. Derivation of Newton’s Second Law

Newton’s Second Law states:

\[ F = ma \]

Using momentum \( p = mv \), we derive:

\[ F = \frac{dp}{dt} \]

For constant mass, this simplifies to \( F = ma \).

4. Derivation of Gravitational Potential Energy

Potential energy \( U \) is derived by integrating force:

\[ U = \int F dx \]

For gravity:

\[ U = mgh \]

5. Conservation of Mechanical Energy

The principle of conservation of mechanical energy states that the total mechanical energy (sum of kinetic and potential energy) remains constant in the absence of external forces:

\[ KE + PE = \, constant \]

Expanding this, we get:

\[ \frac{1}{2} m v^2 + mgh = constant \]

This principle is useful in solving problems related to free fall, pendulums, and roller coasters.

6. Derivation of Bernoulli’s Equation

Bernoulli’s principle states that in a streamlined flow of an incompressible fluid, the sum of pressure energy, kinetic energy, and potential energy per unit volume remains constant:

\[ P + \frac{1}{2} \rho v^2 + \rho gh = constant \]

This equation is derived from the work-energy principle applied to fluid flow.

7. Ohm’s Law and Power Formula

Ohm’s Law states:

\[ V = IR \]

where \( V \) is voltage, \( I \) is current, and \( R \) is resistance.

The power dissipated in an electrical circuit is derived as:

\[ P = VI \]

Substituting Ohm’s Law:

\[ P = I^2 R = \frac{V^2}{R} \]

8. Example Calculation: Motion of a Falling Object

Given:

\( u = 0 \) (initial velocity)
\( a = 9.8 m/s^2 \) (acceleration due to gravity)
\( t = 5 \) seconds

Using \( v = u + at \):

\[ v = 0 + (9.8)(5) = 49 m/s \]

Now, finding displacement using \( s = ut + \frac{1}{2}at^2 \):

\[ s = 0 + \frac{1}{2} (9.8) (5^2) \]

\[ s = \frac{1}{2} (9.8) (25) = 122.5 m \]

9. Example Calculation: Electrical Power

Given a resistor with \( R = 10 \Omega \) and a current of \( 5A \), we calculate power:

\[ P = I^2 R = (5)^2 (10) = 250W \]

Conclusion

Derived physics formulas help solve complex problems efficiently. Understanding derivations provides deeper insight into physical principles. These derivations and equations are essential in engineering, mechanics, and various scientific applications.