Question

Derive
(1) =u+at
(ii) v2 -u? = 2 as by calculus method.​

Answer 1

(1)v = u +at

Here, u is inital velocity,v is final velocity,a is acceleration,s is displacement and t is time interval.

Proof:

Let us consider an object is moving with initial velocity u. After a time interval t its final velocity be v and the object gain an acceleration a.

we have,

Acceleration = Final velocity - Initial velocity/time taken

a = v -u/t

at = v - u

v = u + at

(ii)v^2 -u^2 = 2as

The second equation of motion is

s = ut +1/2at^2

By substituting the value of t from eqn.(i) in eqn.(ii),we get

s = u (v-u/a) +1/2a(v-u/a)^2

s = u(v-u)/a +1/2a(v-u)^2/a^2

s = (uv- u)^2/a + (v-u)^2/2a

s = 2(uv - u)^2 + (v - u) ^2/2a

2as = 2(uv - u^2) + (v - u)^2

2as = 2uv - 2u^2 +v^2 +u^2 -2uv

2as = -u^2 + v^2

v^2 = u^2 +2as

.v^2 - u^2 = 2as

Hence, formula is derived

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