Question

Explain 3 equations of Motions using Calculus Method..
With detailed explanation

Answer 1

Explanation:

There are three equations of motion that can be used to derive components such as displacement(s), velocity (initial and final), time(t) and acceleration(a). The following are the three equation of motion: First Equation of Motion : v=u+at. ... Third Equation of Motion : v^2=u^2+2as.

Answer 2

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Let us Explain the 3 equations of motions using integration method.

$\huge \tt\underline{\underline{1st\: Equation\: of \:Motion :}} \green \checkmark$

Let the small change in velocity be dv.

Since acceleration is uniform,

➠ a = $\dfrac{dv}{dt}$

➠ dv = adt

Let us apply Integration on both sides

$\huge\sf\red{ \int\limits^v_u dv } \: = \int\limits^t_0 {a} \, dt$

as "a" is constant acceleration

$\huge\sf\red{\int\limits^v_u dx } \: = a \int \limits^t_0 dt$

$\implies v -u \: = a(t-0)$

Hence 1st equation of motion is given as

$\huge\star \sf{\underbrace\green{v\: = u+at }}$

$\huge \tt\underline{\underline{2nd\: Equation\: of \:Motion :}} \green \checkmark$

dx = vdt

Let us apply integration on both sides

$\huge\bf\ \blue{\int\limits^S_0 dx } \: = \int\ vdt$

v can substituted as u+at

$\huge\bf\ \blue{s\:= \int\ {(u+at)} \,. dt }$

$s \: = \int\limits^t_0 {u} \, dt + \int\limits^t_0 {at} \, dt$

$s \: = u\int\limits^t_0 dt + a\int\limits^t_0 {t} \, dt$

$s \: = u(t-0) + \dfrac{at^{2} }{2}$

$\huge\blue{ \boxed{S\: = ut+\dfrac{1}{2}at^{2} }}$

$\huge \tt\underline{\underline{3rd\: Equation\: of \:Motion :}} \green \checkmark$

As we already know that

$a =\dfrac{dv}{dt}$

$\implies a\: = \dfrac{dv}{dt} \times \dfrac{dx}{dx}$

$\implies a\: = \dfrac{dv}{dx} \times \dfrac{dx}{dt}$

$\implies a\: = \dfrac{dv}{dx} \times v$

$\implies a\: = \dfrac{vdv}{dx}$

$\implies adx\: = {vdv}$

Apply integral limits

$\huge\bf\ \blue{\int\limits^S_0 adx } \: = \int\limits^v_0 vdv$

$as\, = \dfrac{v^{2} -u^{2} }{2}$

Hence 3rd equation of motion is given as,

$\huge \boxed{\boxed {\red{v^{2}-u^{2}\: = 2as }}}$

Hence proved

$Hope\: It\: Helps$