The Kinematic Equations Explained

The Kinematic Equations Applied

The kinematic equations describe the motion of an object in classical mechanics. In the simple case of uniformly accelerated motion in one dimension, they provide us with useful results for relating the variables involved. These variables include velocity (v), time (t), acceleration (a) and displacements (s) of the object.

So, let us look at the four kinematic equations.

Velocity – Time Relation

How velocity relates to acceleration.
How velocity relates to acceleration. Source.

This relation can be obtained from the definition of acceleration:

In the case of uniform acceleration in one dimension, this reduces to: 

Where
v is the final velocity
u is the initial velocity
Δt is the time interval.

If we take the initial time as 0, we can just write Δt = t as done above

Thus, by rearranging terms, we obtain the first kinematic equation:

First kinematic equation - Velocity - Time Relation

This kinematic equation is the velocity-time relation

Example: If a particle, starting at u = 2 m s-1 accelerates at a = 0.1 m s-2 for t = 20 s, What will be its final velocity?

Applying the velocity-time relation, we get that its final velocity
v = 2 + (0.1)(20)
=> 2 + 2
=> 4 m s-1 

Displacement – Velocity Relation (With Time)

The average velocity vav is just a time-averaged value of the object’s velocity at various times. The average velocity formula is as follows:

In the case of uniformly accelerated motion in one dimension, the average velocity has a simple formula. It is equal to half of the sum of the initial velocity (u) and final velocity (v)

Using the above two equations, we get the result:

Second Kinematic Equation. Displacement - Velocity Relation (With Time)

This kinematic equation is the relationship between displacement and velocity, with time present.

Visualizing the distance covered as the area on the velocity-time graph.
Visualizing the distance covered as the area on the velocity-time graph. (Source)

Example: Consider a particle that starts with velocity u = 3 m s-1 and has final velocity v = 1 m s-1. If it travels for t = 30 s, What is the displacement?

The displacement s covered in the time is
= (3+1/2)(30)
=> 60 m

Displacement – Time Relation

If we substitute the results of the velocity-time relation in the expression for average velocity, we can get rid of final velocity v. 

Then, if we substitute this in the previous relation for velocity and displacement, we obtain:

Third Kinematic Equation.  Displacement - Time Relation

This kinematic equation is the relation between displacement and time. 

A case of free-fall. Note that acceleration is constant.
A case of free-fall. Note that acceleration is constant. (Source)

Example: Consider an object accelerating at a = -10 m s-2 for t = 5 s. Let its initial velocity u = 0. What is the distance travelled?

The equation gives us that the distance travelled is
=> s = (0)(5)+(0.5)(-10)(5^2) = -125 m.

Note that the negative sign indicates that the displacement is opposite to the chosen positive axis. This example can be of an object in free fall due to gravity for 5 s. 

Displacement – Velocity Relation (Without Time)

We can get rid of time in the displacement-velocity relation. For this, note that time can be written as: 

Now we eliminate time by substituting the above in the previous displacement – velocity relation. This gives us: 

This result is often stated as

Fourth Kinematic Equation  Displacement - Velocity Relation (Without Time)

This kinematic equation is the displacement-velocity relation. 

Example: Consider a car with initial velocity 20 m s-1 and final velocity 0 m s-1. Let it come to rest in a distance s = 200 m. What is the acceleration?

We can then calculate the acceleration as a = (02 – 202)/200 = – 2 m s-2

Uses

The kinematic equations have a wide variety of uses in cases where a particle accelerates at a uniform rate. In classical mechanics, this is equivalent to the case where the particle is acted upon by a constant force. A common case where this can be used is in studying a freely falling object. 

Further, we can extend these equations to higher dimensions, if we take suitable components of velocity, acceleration and so on.

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