(B) Given the velocity $v = \frac{dx}{dt} = \alpha \sqrt{x}$.Separating the variables,we get $\frac{dx}{\sqrt{x}} = \alpha dt$.Integrating both sides,

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(B) Given the velocity $v = \frac{dx}{dt} = \alpha \sqrt{x}$.Separating the variables,we get $\frac{dx}{\sqrt{x}} = \alpha dt$.Integrating both sides,$\int_{0}^{x} x^{-1/2} dx = \int_{0}^{t} \alpha dt$.This yields $[2x^{1/2}]_{0}^{x} = \alpha t$.So,$2\sqrt{x} = \alpha t$,which implies $\sqrt{x} = \frac{\alpha}{2} t$.Squaring both sides,$x = \frac{\alpha^2}{4} t^2$.Therefore,the displacement $x$ is proportional to $t^2$.

Class 11 Physics Motion in Straight Line Uniformly Accelerated Motion

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$A$ particle located at $x = 0$ at time $t = 0$ starts moving along the positive $X$-direction with a velocity $v$ that varies as $v = \alpha \sqrt{x}$. The displacement of the particle with time is proportional to

A
$t^3$
B
$t^2$
C
$t$
D
$t^{1/2}$

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Class 11 Questions

Class 11

Physics Questions

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Motion in Straight Line

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Uniformly Accelerated Motion

The initial velocity of a particle is $u$ (at $t = 0$) and the acceleration is given by $a(t) = at$. Which of the following relations is valid?

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The position of a particle as a function of time $t$ is given by $x(t) = at + bt^2 - ct^3$,where $a, b,$ and $c$ are constants. When the particle attains zero acceleration,its velocity will be:

DifficultJEE MAIN 2019

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Derive the equations of uniformly accelerated motion by the graphical method.

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$A$ car moving with a velocity of $20 \,m \,s^{-1}$ is stopped in a distance of $40 \,m$. If the same car is travelling at double the velocity, the distance travelled by it for the same retardation is (in $\,m$)

MediumKCET 2017

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An alpha particle enters a hollow tube of $4 \, m$ length with an initial speed of $1 \, km/s$. It is accelerated in the tube and comes out of it with a speed of $9 \, km/s$. The time for which it remains inside the tube is

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$A$ particle located at $x = 0$ at time $t = 0$ starts moving along the positive $X$-direction with a velocity $v$ that varies as $v = \alpha \sqrt{x}$. The displacement of the particle with time is proportional to

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