Question

Derive the formula :-
v = u + at
using differentiation.
(Acceleration is constant)

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Answer 1

CORRECT QUESTION :-

Derive the formula :-

v = u + at

using integration.

(Acceleration is constant)

SOLUTION :-

GIVEN :-

$\overrightarrow{a}$ is constant.

TO DERIVE :-

v = u + at

DERIVATION :-

We know,

$\sf{\displaystyle{\overrightarrow{a}=\frac{d \overrightarrow{v}}{dt} }}$

$\sf{\displaystyle{\implies \overrightarrow{a}\ dt=d \overrightarrow{v} }}\\\\\sf{\displaystyle{\implies \int{\overrightarrow{a}dt}=\int{d \overrightarrow{v} }}$

$\sf{\displaystyle{\implies \verrightarrow{a} \int\limits^t_0 {dt} = \int\limits^{\overrightarrow{v}_f} _{\overrightarrow{v}_i}d \overrightarrow{v}}}$

$\sf{\displaystyle{\implies \overrightarrow{a} |t|^t_0=|\overrightarrow{v}|^{\overrightarrow{v}_f}_{\overrightarrow{v}_i}}}$

$\sf{\displaystyle{\implies \overrightarrow{a}(t-0)=\overrightarrow{v}_f-\overrightarrow{v}_i}}$

$\boxed{\sf{\displaystyle{\implies \overrightarrow{v}_f=\overrightarrow{v}_i+\overrightarrow{a}t}}}$

Or we can write it as :

$\underline{\underline{\boxed{\boxed{\sf{\displaystyle{v = u + at}}}}}}$

$\rule{200}{2}$

Here,

$\bigstar\ \sf{v_f=Final\ velocity}\\\\\bigstar\ \sf{v_i=Initial\ velocity}$

Answer 2

To Derive :

• v = u + at using calculus method

Derivation Using Calculus :

We know , for acceleration 'a' velocity 'v' and time 't' the expression is given by

$\sf \implies a = \dfrac{dv}{dt} \\\\ \sf \implies dv = a.dt ........ (1)$

Let a body has velocity 'u' initially (t= 0) and finally (t = ts) aquires velocity 'v'

Now integrating the equation (1) we have :

$\sf \implies \int_{u}^{v} dv= a\int_{0}^{t} dt \\\\ \sf \implies [ v ]_{u}^{v} = a [ t ]_{0}^{t} \\\\ \sf \implies v - u = a (t - 0) \\\\ \sf \implies v - u = at \\\\ \sf \implies v = u + at$