Kinematics Formula: Definition, Derivation & Example

Kinematics is the study of motion of any particular point or object with respect to space and time. It is a branch of classical mechanics. Kinematics deals with the study of motion of any points, objects and groups of objects while ignoring the causes of motion. Kinematics is also referred to as “Geometry of Motion” by many physicists.

Kinetic Energy of an object is the energy that the body possesses due to its motion. It is defined as the work needed to accelerate a body of given mass from rest to a stated velocity.

In order to describe motion, Kinematics focuses on various geometric objects and differential properties including velocity and acceleration. Kinematics is used in astrophysics, mechanical engineering, robotics and biomechanics.

Kinematic equations are a set of equations or formula that derive one of the five kinematic variables namely,

• Displacement (D),
• Initial Velocity (u),
• Final Velocity (v),
• Time Interval (t),
• Constant Acceleration (a)

when other variables are provided. These equations define motion either at constant velocity or at constant acceleration. Since such is the case, we cannot use the equations if either of the two is variable or changing.

Read more: Period Motion

KeyTerms: Kinematic, Velocity, Acceleration, Displacement, Time, Mass, Motion.

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Kinematic Formulas

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The Kinematic Equations of motion are as follows:

1. v = u + at
2. v² = u² + 2aD
3. D = ut + \(\frac{at^2}{2}\)
4. D = \((\frac{v + u}{2}) t\)

Read more:

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Kinematic Formula Derivations

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Formula 1: v = u + at

Derivation:

We know that Acceleration of a body is the rate of change of velocity of that body.

Now, let us assume that a body has mass “m” and initial velocity “u”. After some time “t” its velocity changes to “v” which is its final velocity due to uniform acceleration “a”.

We know that,

\(Acceleration = \frac{Final Velocity - Initial Velocity}{Time Taken}\)

Or, a = \(\frac{v - u}{t}\)

Or, at = v - u

Therefore, v = u + at [Derived]

Formula 2: v² = u² + 2aD

Derivation:

Let us assume, Displacement be “D”

From the first equation, we know that, v= u + at . Therefore, we can write it as, 

t=(v - u)/ a

Also, we know that, Displacement= Average Velocity Time

Hence, for constant acceleration we can write,

Average Velocity = (Final Velocity + Initial Velocity)/2

= (v + u)/2

Therefore, Displacement D= \([\frac{v + u}{2}] \times [\frac{v-u}{a}]\)

Or, D = (v² – u²)/2a

Or, 2aD = v² – u²

Or, v² = u² + 2aD [Derived]

Formula 3: D = ut + \(\frac{at^2}{2}\)

Derivation:

We know that,

Displacement = Average Velocity x Time

Also,

Average Velocity = (u+v) / 2

Displacement D = {\({\frac{u+v}{2}}\)} x t

From the first equation = u + at ,

We get,

D = \(\frac{u + u + at}{2} \times t\)

Or, D = \(\frac{2u + at}{2} \times t\)

Or, D = (2ut + at²) / 2

Or, D = \(\frac{2ut}{2} + at^2/2\)

Therefore, D = ut + \(\frac{at^2}{2}\) [Derived]

Formula 4: D = \((\frac{v + u}{2})t\)

Derivation:

Consider the above velocity – time graph with constant acceleration. The slope of velocity touching the y-axis is represented as acceleration and the area below the slope is displacement of the object “D”.

The base of triangle is “t” the height of the triangle is “v – u" 

So, the area of the triangle will be = ½ t (v – u)

When we add the areas of rectangle and triangle, we get,

 D = ut + ½ t (v – u)

Simplifying this we get,

 D= ut + ½ vt – ½ ut

Combining initial value terms, we get,

 D = ut + ½ at²

On further simplification, we get,

D= \((\frac{v+u}{2})t\) [Derived]

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Rotational Kinematics

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There is another branch of kinematics equations which deals with the rotational motion of a point or an object besides the Transitional or Linear Kinematic equations as discussed earlier that involves kinematics of bodies in linear motion. Displacement is replaced by a change in angle.

• Initial and final velocities are replaced by initial and final angular velocity.
• Acceleration is replaced by angular acceleration
• Time is the only constant

Rotation Kinematics 

The equations of Rotational Kinematics are:

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Things to remember

• The point or object always moves in a straight line and in the same direction.
• The magnitude of velocity is equal to speed.
• The average velocity is equal to instantaneous velocity.
• Since velocity is constant, acceleration is 0 and hence net force or resultant force is also 0.
• Acceleration is a vector quantity.
• If the value of acceleration is negative, it is called Retardation.
• Velocity is a vector quantity and speed is a scalar quantity.
• Units of speed and velocity are “meter per second” or “ms^-1”.