Question

establish v=u+at by calculus method

Answer 1

Explanation:

Your answer refer in Attachment

Thank you

Answer 2

Answer:

$\bullet \; \large{\tt v = u + at}$

To Prove:

• Kinematics First Equation.

Explanation:

$\rule{300}{1.5}$

Acceleration:

It is defined as rate of change of velocity w.r.t time.

S.I units: m/s²

Expression:

$\qquad \large{\tt a = \dfrac{\Delta V}{\Delta t}}$

Talking over a small interval of time,

$\longmapsto \large{\tt a = \dfrac{dv}{dt}}$

Rearranging,

$\longmapsto \large{\tt dv = a.dt}$

Taking over a large time.

$\longmapsto \large{\tt \displaystyle\int \tt dv = \displaystyle\int \tt a.dt}$

Applying Limits,

$\longmapsto \large{\tt \displaystyle\int\limits_{u}^{v} \tt dv = \displaystyle\int\limits_{0}^{t} \tt a.dt}$

$\longmapsto \large{\tt \displaystyle\int\limits_{u}^{v} \tt dv = a \displaystyle\int\limits_{0}^{t} \tt dt}$

Simplifying,gives,

$\longmapsto \large{\tt \bigg[v \bigg]^v_u = a \bigg[t \bigg]^t_0}$

$\longmapsto \large{\tt v - u = at}$

$\longmapsto \large{\underline{\boxed{\red{\tt v = u + at}}}}$

Hence Derived !!

$\rule{300}{1.5}$

$\rule{300}{1.5}$

Special Cases:

1. If final velocity is zero i.e. ( v = 0) then equation becomes, u = - at.
2. If initial velocity is zero i.e. ( u = 0) then equation becomes, v = at.
3. If Acceleration is zero i.e. ( a = 0) then equation becomes, v = u.
4. If time is zero i.e. ( t = 0) then equation becomes, v = u.

$\rule{300}{1.5}$