The equation of motion of a body moving with constant acceleration along a straight line can be derived from Newton's second law of motion, which states that the acceleration of an object is proportional to the force acting on it and inversely proportional to its mass.
Let's consider a body moving with constant acceleration along a straight line and let's call the initial velocity of the body at time t=0 as v0, the final velocity of the body at time t as v, the constant acceleration of the body as a, the time elapsed as t and the displacement of the body as x.
The first equation we can derive is the velocity equation:
v = v0 + at
This equation states that the final velocity of the body at time t is equal to its initial velocity plus the product of its acceleration and time. This means that if the body was initially at rest (v0 = 0), its velocity will increase linearly with time as long as the acceleration remains constant. If the acceleration is positive, the velocity of the body will increase and if the acceleration is negative, the velocity of the body will decrease.
Next, we can derive the displacement equation:
x = x0 + v0t + (1/2)at^2 (Check the image for the exact formula)
This equation states that the final displacement of the body at time t is equal to its initial displacement plus the product of its initial velocity and time plus half of the product of its acceleration and the square of time. This equation takes into account the change in velocity of the body over time and the distance traveled during that change.