Derivation of Equations of motion for a uniformly accelerated object.
First equation of motion :
v = u + atWe all know that,
Acceleration = Final velocity − Initial velocity / Time taken
a = (v − u) / t
From this, we can derive our first equations of motion,
at = v − u
v = u + at ----------------- 1
∴ we derived our first equation of motion, i.e, v = u + at
Second equation of motion:
The second equation of motion is S = ut + ½(at²)We can derive it now.
We know that,
average velocity = (u + v) / 2 ------------------- 2
We also know that,
average velocity = Displacement (S) / Time taken (t) -------------------- 3
Now we can equate the equation 2 and 3.
( u + v ) / 2 = S / t ------------------------ 4
[ (u + v) / 2) ] × t = S --------------------------- 5
S = [ (u + v) / 2) ] × t -------------------------- 6
It is now the time to substitute the equation 1 in the equation 6
So we obtain,
S = [ (u + u + at) / 2) ] × t
S = [ ( 2u + at ) / 2 ] × t
S = [ ( 2u / 2) + (at /2 ) ] × t
S = [ u + ½(at) ] × t
S = ut + ½(at²)
We derived our second equation of motion
Third equation of motion:
v² = u² + 2aSIn equation 5 we can substitute time instead of substituting the velocity to derive the third equation of motion.
[ (u + v) / 2) ] × t = S
From equation 1, we should derive the time,
t = (v − u) / a
Substitute,
S = [ (u + v) / 2) ] × [ ( v − u) / a)]
S = (v² − u²) / 2a
2aS = v² − u²
2aS = v² − u²
2aS + u² = v²
v² = u² + 2aS
Therefore we derived three equation of motion.
