Step-by-Step Derivation of Equations of Motion for Students

Understanding the Derivation Of Equation Of Motion is essential in Physics, as it links velocity, acceleration, and displacement for objects moving with uniform acceleration. This topic covers various derivation methods, such as graphical, algebraic, and calculus, and is fundamental for students in classes 9, 11, and beyond. Read on to master step-by-step derivations, key formulas, and practical examples.

What Are Equations of Motion?

Equations of motion describe how the position, velocity, and acceleration of an object change over time when it moves with uniform acceleration. These relationships allow us to calculate unknown parameters—such as how far a car travels in a certain time, its speed at a particular moment, or the time taken to stop. Whether in class 9, class 11 calculus, or advanced physics, mastering these equations is crucial.

The equations are:

• First equation of motion: relates velocity and time
• Second equation of motion: relates displacement and time
• Third equation of motion: relates velocity and displacement

For a deeper dive into related topics such as velocity, acceleration, or graph-based motion, explore our other guides.

Key Formulas for Equations of Motion

First Equation of Motion (Velocity-Time): $v = u + at$

Second Equation of Motion (Displacement-Time): $s = ut + \frac{1}{2} a t^2$

Third Equation of Motion (Velocity-Displacement): $v^2 = u^2 + 2a s$

Here, $u$ is initial velocity, $v$ is final velocity, $a$ is uniform acceleration, $t$ is time, and $s$ is displacement.

Derivation Of Equation Of Motion: Step-by-Step Methods

Let’s explore the derivation of equation of motion using three main techniques: the algebraic method, the graphical method (suitable for class 9), and the calculus/differentiation method (class 11 and 12).

A. Derivation of First Equation of Motion (Algebraic Method)

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

This is the first equation of motion, linking final velocity, initial velocity, acceleration, and time.

B. Derivation of Second Equation of Motion (Algebraic Method)

1. Displacement $s$ is average velocity $\times$ time. Average velocity = $\frac{u + v}{2}$
2. From above, $v = u + at$ so average velocity is $\frac{u + (u + at)}{2} = u + \frac{1}{2}at$
3. $s = (u + \frac{1}{2}at)t = ut + \frac{1}{2}at^2$

This second equation is useful for finding displacement without knowing the final velocity.

C. Derivation of Third Equation of Motion (Algebraic Method)

1. From first equation: $v = u + at$ → $t = \frac{v-u}{a}$
2. From second equation: $s = ut + \frac{1}{2} a t^2$
3. Substitute value of $t$ from step 1 into step 2:
4. After simplification: $v^2 = u^2 + 2as$

This third equation relates speed and displacement, without time variable.

D. Derivation of Equation of Motion by Graphical Method (Class 9 Approach)

The graphical method involves using the velocity-time graph to demonstrate these equations. For the derivation of equation of motion class 9 graphical method, consider:

1. Plot initial velocity at point $A$, final velocity at $B$ over time $t$
2. Acceleration = slope of $v-t$ graph = $\frac{v-u}{t}$, deducing $v = u+ at$
3. Area under $v-t$ graph gives displacement, which leads to $s = ut + \frac{1}{2}at^2$

For further examples with diagrams, review our guide to equations of motion by graphical method.

E. Derivation of Equation of Motion by Differentiation and Calculus (Class 11/12 Approach)

The derivation of equation of motion class 11 calculus method uses differentiation/integration for deeper insight:

1. $a = \frac{dv}{dt}$ (definition of acceleration)
2. Integrate both sides: $\int_{u}^{v} dv = \int_{0}^{t} a\,dt$
3. $v - u = a t \implies v = u + at$
4. For displacement, $v = \frac{ds}{dt}$, $a = \frac{dv}{dt}$, chain rule gives $a = v \frac{dv}{ds}$
5. Integrate: $\int_{u}^{v} v\,dv = \int_{0}^{s} a\,ds$ → $\frac{v^2 - u^2}{2} = a s$ → $v^2 = u^2 + 2 a s$

This differentiation method matches the derivation of equation of motion by differentiation method in senior Grades.

Summary Table: Derivation of First, Second, and Third Equations of Motion

This table condenses the derivation of equation of motion first, second, and third in one place for easy revision and reference.

Practical Applications and Numerical Example

Let’s apply the equations to a simple problem: A car starts from rest ($u = 0$) and accelerates at $2\,\text{m/s}^2$ for $5$ seconds. Find displacement and final velocity.

1. Final velocity: $v = 0 + (2)(5) = 10 \ \text{m/s}$
2. Displacement: $s = 0 \times 5 + \frac{1}{2} (2) (5)^2 = 25\ \text{m}$

Such calculations are vital in physics, engineering, and real life. Learn more about related kinematic and dynamic effects like force, average speed and velocity, and uniform and non-uniform motion to develop your problem-solving skills further.

The derivation of equation of motion in physics is not just a classroom exercise. It is widely used in vehicle motion, projectile trajectories, engineering dynamics, and even in sports science to model movement.

Conclusion: Mastery of Derivation Of Equation Of Motion

Mastering the Derivation Of Equation Of Motion—whether by graphical, algebraic, or calculus methods—builds a foundation for all motion analysis in physics. Use these equations of motion to solve a variety of problems from class 9 to class 12 and beyond. To sharpen your understanding, revisit topics like distance and displacement or explore advanced concepts using our physics formulas for class 9 resource library.