Equations of motion with constant acceleration | POINT classical mechanics

 The 3 equations of motion:

Motion is an integral part of our daily experience, whether we observe a car speeding down the road or an athlete sprinting on a track. In physics, understanding motion under constant acceleration is made accessible through three fundamental equations. These equations provide a clear and concise framework that links initial velocity, final velocity, acceleration, time, and displacement. Designed for simplicity, they allow us to predict how objects move without the need for advanced mathematical tools. This article focuses on the elegant relationships among these variables, emphasizing their practical significance and the straightforward logic behind them. By concentrating solely on the core principles of these kinematic equations, we invite readers to appreciate the inherent order and predictability of motion in our everyday world.

You can describe body's movement in straight line with velocity changing from initial velocity (Vi) to final velocity (Vf) to cover displacement (d) in a certain time (t) with constant acceleration (a) by 3 equation called: Equations of motion with constant acceleration:

Considering the beginning of the movement at time = 0 Then:

Multiply both sides of the equation by (t):

The graph of the first equation of motion at:

The second equation of motion (displacement-time equation):

The average velocity (v) of a body moving a displacement (d) in a time (t) is determined from the equation:

Since the body is moving with constant acceleration, the average velocity can be calculated from the equation:

from 1 , 2

Substituting (Vf) from the first equation of motion:

Multiply both sides of first equation of motion:

The graph of the second equation of motion at v = 0 can be drawn as follows:

Derive the second equation of motion graphically.

(displacement = velocity × time)                                              from the graph: displacement (d) = the area under the curve (velocity-time)

*The area under the curve is divided into a rectangle and a triangle:

Area of rectangle (A1) = length × width

Area of triangle (A2) = 1/2 (Base length x height)

By adding the areas (Equations 1, 2)

The third equation of motion (displacements-velocity equation):

The displacement of moving body determined from this equation :

(And from the first equation of motion)

Substituting from the 2 and 3 equation in the 1 equation:

The graph of the third equation of motion at:

The opposite triangle is used to solve equations of motion problems, where the number written between the two quantities (known and unknown) indicates the number of the equation of motion used in the solution:

The three equations of motion form a succinct yet powerful framework within kinematics. They encapsulate the essential relationships between an object's initial and final velocities, the constant acceleration it experiences, and the displacement covered over time. This analytical approach not only deepens our theoretical understanding but also provides a versatile set of tools for solving a wide range of problems in physics. By focusing exclusively on the derivations and intrinsic properties of these equations, we appreciate their logical coherence and mathematical beauty. Their enduring relevance in physics is a testament to the fundamental nature of these relationships, which continue to inspire both academic inquiry and practical applications in more abstract settings. As we reflect on these equations, we recognize that their simplicity belies a profound insight into the mechanics of motion—a cornerstone of classical physics that remains as relevant today as it was in the pioneering days of scientific discovery.