Question

Derive following equations for a uniformly accelerated motion :

(i) v=u+at

(ii) $S=ut+\frac{1}{2}a{t}^2$

(iii) $v^2=u^2+2aS$

where the symbols have their usual meanings.

Answer 1

(1) First equation of Motion:

V = u + at
soln.
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 we know that:
Acceleration = change in velocity/Time taken

=> Acceleration = Final velocity-Initial velocity / time taken

=> a = (v-u) /t
=>at = v-u

=>v = u + at
This is the first equation of motion.
—————————————-
(2) Second equation of motion:

sol.
Let the distance travelled by the body be “s”.
We know that
Distance = Average velocity X Time

Also, Average velocity = (u+v)/2

.: Distance (t) = …….eq.(1)

Again we know that:
v = u + at
substituting this value of “v” in eq.(1), we get

This is the 2nd equation of motion.
……………………………………………………………
(3) Third equation of Motion

sol.
We know that
V = u + at
=> v-u = at
or t = (v-u)/a ………..eq.(2)

Also we know that
Distance = average velocity X Time
.

This is the third equation of motion