Equations of Motion: Basic Concepts, Derivations and Examples

Introduction:

The equations of motion also known as kinematic equations are the equations that are used to describe the motion of a particle or object moving in 1D, 2D or 3D space. Be the motion be uniform or non-uniform, accelerated or non-accelerated, mathematical equations relating the different parameters of motion can be framed to represent, understand and describe the motion. In this article we are going to talk about the four basic equations of motion and their derivation which are applicable specifically for uniformly accelerated motion.

The Four Basic Equations of Motion:

If an object is observed by some observer to make a displacement s in a time interval t with initial velocity u and final velocity v and if the motion was uniformly accelerated in that interval of time at the rate of a, then these parameters of motion (i.e. u, v, a, t & s) are interrelated by some mathematical equations. These are known as the basic equations of motion and are four in number :

Understand the Exact Meaning of Notations Used:

Here the most important point to note is to understand the exact meaning of the notations used. Many get confused at this point which results in wrong use of these equations. In the equations u and v stand for initial and final velocities (and not speeds) respectively. Similarly a represents the resultant acceleration vector during the motion (and not just the tangential component of net acceleration which represents the time rate of speed change). s is the displacement and not necessarily the distance traversed. Also the motion should be uniformly accelerated i.e. the acceleration a must not be a function of time. It is also noteworthy that each of the frame dependent quantities u, v, a, & s  must be measured with respect to the same frame of reference. Out of the these five parameters (i.e. u, v, a, t & s) if any three is known, the fourth can easily be obtained using one of the equations from the above set.

The equations of uniformly accelerated motion written above are in their most general form and are applicable for motions taking place in 1D and 2D as well provided the above mentioned conditions are satisfied. It is again a common mis-conception that these equations of motion are only valid for rectilinearly moving objects. For instance, we can use these formulas while dealing with an ideal projectile motion taking place very near to the earth's surface (so that the acceleration due to gravity can approximately be treated as time-independent).

Talking for the special case when the particle moves along a straight line path, the only two possible directions of the vector terms u, v, a, & s associated with such motion can be represented or distinguished simply by using plus(+) and minus(-) signs of algebra eliminating the need to write them with additional arrow sign and thus the kinematic equations take the following simple form:

Now let's proceed to derivations using simple calculus.

Derivation of Equations of Motion

Let's assume that an object makes a displacement s in a time interval t with initial velocity u and final velocity v and the motion was uniformly accelerated in that interval of time at the rate of a with respect to a certain observer or frame. By definitions we know that the velocity is time rate of change of displacement and the acceleration is the time rate of change of velocity. This information is sufficient to carry out the derivation.

Now to get the remaining fourth equation of motion, just eliminate the acceleration term a from the equations 1 and 3. From the first equation of motion, we have