Equations of Motion and Their Derivations

Equations of Motion

Motion is an important concept in the branch of physics. It is defined as the change in the position of a body in a given interval of time with respect to its surroundings. There are many conceptual applications of motion, which can be better understood by applying the concept of motion in the questions.

This article will cover in detail – what is motion, the types of motion, the equations of motion and the derivation of equations of motion. You can also find further information here: equations of motion pdf 


Motion can be defined as the movement of any object in a given time interval with respect to its surroundings. The motion of any object with mass can be described as:

  • Speed 
  • Velocity
  • Acceleration
  • Time
  • Displacement
  • distance

There are various types of motion depending on what kind of force is acting on the object with mass. The different types of motions are described below:

  1. Linear motion: A body moving along a single dimension in a single direction is termed linear motion. It is one of the types of translational motion.
  2. Translational motion: A body moving along a path in any given three dimensions is termed translational motion.
  3. Periodic motion: As the term suggests, Periodic motion is the kind of motion that repeats itself after periodic intervals.
  4. Rotational motion: Rotational motion, as the term suggests, is the kind of motion that moves along the fixed axis, forming a circular path.
  5. Projectile motion: This kind of motion has two types of displacement – horizontal displacement and vertical displacement.
  6. Oscillatory motion: Oscillatory motion should not be confused with periodic motion because of its repetitive nature. This kind of motion is repetitive in a certain time frame. If the motion is mechanical, it is called vibration.
  7. Simple Harmonic motion: This is the kind of motion similar to that of a pendulum. Here, the displacement of an object from the mean position is directly proportional to its restoring force. The restoring force of the object acts in the opposite direction of the direction of motion.

Before we learn about the equations of motion, let’s brush up on the knowledge of the laws of motion. The three laws of motion are described below:

  • First Law: The first law of motion states that the object will remain in the state of inertia or motion unless an external force is applied to the object to change its initial state.
  • Second Law: The second law of motion states that the greater the mass of any object, the greater force required to move that object. It can be identified by the formula F = m * a, where ‘F’ is the external force applied to the object, ‘m’ is the mass of the object and ‘a’ is the acceleration. Here, we can see that force is directly proportional to the mass of the object. 
  • Third Law: The third law of motion states an equal and opposite reaction for every action.

Equations of Motion

Equations of motion and their derivations are the most important and basic topics in the branch of physics. The equations of motion underline the motion of an object in all three dimensions – 1D, 2D and 3D, with respect to the velocity (v and u) or acceleration (a) of the object, its position, at various intervals of time. 

This section will explore the three equations of motion and their derivation by algebraic, calculus and graphical methods.

The equations of motion are used to determine components such as velocity (initial and final velocity), acceleration (a), displacement (s) and time(t):

  • The first equation of motion is v = u + at.
  • The second equation of motion is s = ut + (½) at²
  • The third equation of motion is v² = u² + 2as 

Derivation of First Equation of Motion

To derive the first equation of motion, consider an object moving with uniform acceleration in a straight line. The acceleration is denoted by ‘a’, the initial velocity is denoted by ‘u’, the final velocity is denoted by ‘v’, the time period is denoted by ‘t’, and the distance travelled by that object is denoted by ‘s’.

Derivation of the first equation by the algebraic method:

As we know that the rate of change of velocity is termed as the acceleration of the body. Acceleration is represented as,

a = (v – u) / t

Here, v is the final velocity, u is the initial velocity, and t is the time.

After the rearrangement of the above equation, we get:

 v = u+ at

Derivation of Second Equation of Motion

To derive the second equation of motion, we consider the same variables used in the derivation of the first equation. As we know that the velocity is the rate of change of displacement, 

Velocity (v) = displacement (s) / time (t)

After the rearrangement of this equation, we get

Displacement (s) = velocity (v) * time (t)

If velocity is not constant, we use the average of initial velocity (u) and final velocity (v), 

Displacement (s) = [(u+v) / 2 ] * time (t)

From the first equation of motion, we know that v = u + at. Now substituting the value of v in the equation, we get

Displacement (s) = [( u + (u + at)) / 2] * t

s = [(2u + at) / 2] * t

On simplification of the equation, we get

s = u + ½at²

Derivation of Third Equation of Motion

As we know that displacement is the rate of the position of the object, which is represented as,

Displacement (s) = [(initial velocity + final velocity) / 2] * time

s = [(u + v) / 2] * t

From the first equation of motion we know, v = u + at

After rearranging the equation, we get

t = (v – u) / a

Now, substitute the equation of ‘t’ in the displacement formula:

s = [(v + u) / 2] * [(v – u) / a]

On simplification of the above equation, we get

v² = u² + 2as


Understanding the foundation of any physics concept plays an important role in further exploring the next sections of the same topics. If you are thorough with velocity, displacement, and acceleration concepts, equations’ derivations will come easily into practice. These concepts are necessary under classical mechanics; in other words, they set the foundation for the next level concepts in physics.

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