Question

Derive the first equation of the motion by algebraic method

Answer 1

dv/dt =a
dv=adt
integrating within proper limits
v-u=a(t-0)
v-u=at
v=u+at

Answer 2

Answer:

The first equation of motion is : v = u + at.

Derivation by algebraic method -

Let us consider a uniformly accelerated body that was travelling with initial velocity u, and is travelling with a final velocity v at time t seconds.

We know that, Acceleration is the rate of change of velocity.

Therefore, a = (v - u)/t

⇒ a = (v - u)/t

⇒ at = v - u

Hence, v = u + at.

Let us derive this equation by calculus method -

We know that, a = Δv/Δt = dv/dt.

⇒ dv = a dt

Using integration on both sides of the equation,

$\longrightarrow\tt{ \int\limits^v_u \,dv = \int\limits^0_t a\,dt}$

$\longrightarrow\tt{\int\limits^v_u \,dv = a \int\limits^0_t \,dt}$

$\longrightarrow\tt{ [v]_u^v = a[t]_0^t}$

$\longrightarrow\tt{v - u = a(t-0)}$

$\longrightarrow \tt{v-u=at}$

Therefore, v = u + at.