Question

A particle moving along x -axis has acceleration f, at

time t, given by :–


$f = f_0 \Bigg( 1 -\dfrac{t}{T}\Bigg)$
where $f_0$ and T are

constants. The particle at t = 0 has zero velocity. In
the time interval b/w t = 0 and the ins tant when
f = 0, the particle’s velocity (vx) is ?

​

Answer 1

Answer:

${\bold{\green{{\underline{Given}}}}}$

f = acceleration

f = $f_0(1 - \frac{t}{T} )$

t = 0 , V = 0

_______________________

${\pmb{\underline{\sf{Required \: Solution}}}}$

$: \implies0 = f _{0}(1 - \frac{t}{T} ) \\ : \implies1 - \frac{t}{T} = 0 \\ : \implies \frac{t}{T} = 1 \\ \boxed {\implies \: t \: = \: T}$

$\;\large{\boxed{\bf{\red{(t = T) = ( V_{x})}}}}$

$: \implies \: f = f_{0}(1 - \frac{t}{T} ) \\ \boxed{f = acc {}^{n} = \frac{dv}{dt} } \\ \\ \implies\frac{dv}{dt} = f_{0}(1 - \frac{t}{T} )$

$: \implies\int \limits _{0}^{vx} dv = f _{0} \int \limits_{0}^{T} (1 - \frac{t}{T} )dt \\ \implies \: Vx = \: (f _{0}( \frac{t - t {}^{2} }{2T} )\int\limits_{0}^{T}\\ \implies Vx = \: f _{0}( \frac{T -T {}^{2} }{2 t} ) \\ \\ :\implies f_0 (t - \frac{t}{2})\\ \\ \boxed{Vx = \frac{f _{0}T}{2} }$

Answer 2

${\boxed{\underline{\tt{\orange{Required \: Answer:-}}}}}$

$\displaystyle \sf \large{v = f_0 \frac{T}{2} }$

★SOLUTION:-

In this problem, acceleration is variable.

$: \longrightarrow \displaystyle \sf{f = \frac{dv}{dt} }$

$:\longrightarrow\displaystyle\sf{f_0(1- \frac{t}{T} )}$

At,

• $\sf{t = 0}$

• $\sf{f = f_0}$

• $\sf{T = T}$

• $\sf{f = 0 }$

We have to calculate the velocity of the particle in the time from t = 0 to t= 1s.

$: \longrightarrow \displaystyle \sf{ \frac{dv}{dt} =f_0 (1 - \frac{t}{T} )}$

Integrating both sides:-

$: \longrightarrow \displaystyle \sf{\int dv = \int_0^T f_0 (1- \frac{t}{T} )dt}$

$:\longrightarrow\displaystyle\sf{v = f_0[t- \frac{ {t}^{2} }{2T} ]_0^T}$

$:\longrightarrow\displaystyle\sf{v = f_0[T - \frac{{T}^{2} }{2T}] }$

$:\longrightarrow\displaystyle\sf{v = f_0[T - \frac{T }{2}] }$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: { \boxed{ \huge{ \pink{ \underline{ \underline{\displaystyle{\sf{v = f_0 \frac{T}{2} }}}}}}}}$