The three equations of motion v = u + at ; s = ut + (1/2) at2 and v2 = u2 + 2as can be derived with the help of graphs as described below. Consider the velocity – time graph of a body shown in the below Figure.Velocity–Time graph to derive the equations of motion.

Equations of Motion
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The equations of motion are mathematical representations that describe the motion of an object in terms of its position, velocity, and acceleration. These equations are derived from Newton's laws of motion and can be used to solve a variety of kinematic problems. There are three primary equations of motion:

1. Equation of Motion for Displacement (s):
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The equation of motion for displacement relates the initial velocity (u), time (t), acceleration (a), and displacement (s) of an object. It can be derived by integrating the equation of motion for velocity with respect to time. The equation is as follows:

s = ut + (1/2)at^2

- s: Displacement of the object (in meters)
- u: Initial velocity of the object (in meters per second)
- t: Time taken (in seconds)
- a: Acceleration of the object (in meters per second squared)

2. Equation of Motion for Velocity (v):
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The equation of motion for velocity relates the initial velocity (u), time (t), acceleration (a), and final velocity (v) of an object. It can be derived by rearranging the equation of motion for displacement. The equation is as follows:

v = u + at

- v: Final velocity of the object (in meters per second)

3. Equation of Motion for Acceleration (a):
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The equation of motion for acceleration relates the initial velocity (u), final velocity (v), time (t), and acceleration (a) of an object. It can be derived by rearranging the equation of motion for displacement and substituting the equation of motion for velocity. The equation is as follows:

v^2 = u^2 + 2as

- a: Acceleration of the object (in meters per second squared)

Explanation:
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- These equations of motion are derived from Newton's second law of motion, which states that the acceleration of an object is directly proportional to the net force applied to it and inversely proportional to its mass.
- The equations can be used to solve a variety of kinematic problems, such as calculating the final velocity of an object after a given time period or determining the displacement of an object from its initial position.
- By using these equations, one can analyze the motion of an object and make predictions about its future behavior.
- It is important to note that these equations assume constant acceleration, which may not always be the case in real-world scenarios. However, they provide a good approximation for many situations.
- These equations are fundamental in the field of physics and are widely used in various branches of science and engineering to describe and analyze the motion of objects.