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PART - 1 CH - 2 Motion in a Straight Line — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysicsPART - 1 CH - 2 Motion in a Straight Line2 Marks
Question
If the initial velocity of a particle is $u$ and the acceleration in the same direction changes with time as $\alpha t$, then what will be the velocity at time $t$ ?
✓

Answer

 Here the acceleration is changing. Thus Newton's equation cannot be used.
$\begin{aligned}
\therefore \text { Acceleration } a & =\frac{dv}{dt}=\alpha t \\
dv & =\alpha t dt .......(1)
\end{aligned}$
Integration of equation (1)
$\begin{aligned}
\int dv & =\int \alpha tdt \\
v & =\frac{\alpha t^2}{2}+c
.......(2)\end{aligned}$
Here c is integration constant.
Putting $v=u$ at $t =0$
$u=0+c \quad \therefore c=u$
From eqn.(2)
$ \begin{aligned}
v & =\frac{\alpha t^2}{2}+u \\
v & =u+\frac{1}{2} \alpha t^2........(3)
\end{aligned}$

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