Definition of Equations of Motion

Equations of motion, in physics, are defined as equations that describe the behaviour of a physical system in terms of its motion as a function of time.

There are three equations of motion that can be used to derive components such as displacement(s), velocity (initial and final), time(t) and acceleration(a). The following are the three equation of motion:

First Equation of Motion : v=u+at

Second Equation of Motion : s=ut+1/2 at2

Third Equation of Motion : v2=u2+2as

DERIVATION OF FIRST EQUATION OF MOTION ( V = u+at )

Consider a body of mass “m” having initial velocity “u”. Let after time “t” its final velocity becomes “v” due to uniform acceleration “a”.

Now as we know that,

Acceleration = change in velocity/Time taken Acceleration = Final velocity-Initial velocity / time taken

a = v-u/t

=> at = v-u

or v = u + at

This is the first equation of motion.

DERIVATION OF SECOND EQUATION OF MOTION (S=ut+1/2 at2 )

Consider a body of mass “m” having initial velocity “u”. Let after time “t” its final velocity becomes “v” due to uniform acceleration “a”. Let the distance travelled by the body be “s”.

Now As we know that,

Distance = Average velocity X Time

Also, Average velocity = u+v/2

Distance (s) = ( u+v ) *t/2 …….eq.(1)

Again we know that, v = u + at

By substituting this value of “v” in eq.(1), we get

Distance(s) = (u+u+at) *t/2

=> s = (2u+at)* t/2

=> s = 2ut/2+at2/2

=>s = ut+ at2/2

or s = ut + 1/2 at2

This is the second equation of motion.

DERIVATION OF THIRD EQUATION OF MOTION (V2 = u2+2as)

Consider a body of mass “m” having initial velocity “u”. Let after time “t” its final velocity becomes “v” due to uniform acceleration “a”. Let the distance travelled by body be “s”.

Displacement=(Initial Velocity+Final Velocity)×t/2

Substituting the standard notations, the above equation becomes

s=(u+v)×t/2

From the first equation of motion, we know that

v=u+at

Rearranging the above formula, we get

t=v−u/a

Substituting the value of ‘t’ in the displacement formula, we get

s=(v+u)/2 *(v−u)/a

s=(v2 - u2)/2a

2as= v2 - u2

Rearranging, we get

v2 – u2= 2as

This is the third equation of motion.

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