(B) Given that $v = ky^\beta$, where $k$ is a constant. Since $v = \frac{dy}{dt}$, we have $\frac{dy}{dt} = ky^\beta$, which implies $y^{-\beta} dy =

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(B) Given that $v = ky^\beta$, where $k$ is a constant. Since $v = \frac{dy}{dt}$, we have $\frac{dy}{dt} = ky^\beta$, which implies $y^{-\beta} dy = k dt$. Integrating both sides from $0$ to $L$ for distance and $0$ to $T$ for time: $\int_0^L y^{-\beta} dy = \int_0^T k dt$. This gives $\frac{L^{1-\beta}}{1-\beta} = kT$, so $T = \frac{L^{1-\beta}}{k(1-\beta)}$. The average velocity is $\langle v \rangle = \frac{L}{T} = \frac{L}{L^{1-\beta} / (k(1-\beta))} = k(1-\beta) L^\beta$. We are given $\langle v \rangle \propto L^{1/3}$, so comparing the exponents of $L$, we get $\beta = 1/3$.

Class 11 Physics Motion in Straight Line Mix Examples-Motion in Straight Line

Medium

$A$ car moves in the positive $Y$-direction with velocity $v$ proportional to the distance travelled $y$ as $v(y) \propto y^\beta$, where $\beta$ is a positive constant. The car covers a distance $L$ with average velocity $\langle v \rangle$ proportional to $L$ as $\langle v \rangle \propto L^{1/3}$. The constant $\beta$ is given as:

TS EAMCET 2018 All TS EAMCET

A
$1/4$
B
$1/3$
C
$2/3$
D
$1/2$

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Motion in Straight Line

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Mix Examples-Motion in Straight Line

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Two buses $P$ and $Q$ start from a point at the same time and move in a straight line and their positions are represented by $X_{P}(t) = \alpha t + \beta t^{2}$ and $X_{Q}(t) = ft - t^{2}$. At what time,do both the buses have the same velocity?

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When the velocity of a body is variable,then:

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$A$ ball is dropped vertically from a height $d$ above the ground. It hits the ground and bounces up vertically to a height $d/2$. Neglecting subsequent motion and air resistance,its velocity $v$ varies with the height $h$ above the ground as:

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$A$ train starts from rest from a station with acceleration $0.2 \, m/s^2$ on a straight track and then comes to rest after attaining maximum speed at another station due to retardation $0.4 \, m/s^2$. If the total time spent is half an hour,then the distance between the two stations is [Neglect the length of the train]. (in $, km$)

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The displacement of a particle moving in a straight line depends on time as $x = \alpha t^3 + \beta t^2 + \gamma t + \delta$. The ratio of initial acceleration to its initial velocity depends on:

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Two buses $P$ and $Q$ start from a point at the same time and move in a straight line and their positions are represented by $X_{P}(t) = \alpha t + \beta t^{2}$ and $X_{Q}(t) = ft - t^{2}$. At what time,do both the buses have the same velocity?

When the velocity of a body is variable,then:

$A$ ball is dropped vertically from a height $d$ above the ground. It hits the ground and bounces up vertically to a height $d/2$. Neglecting subsequent motion and air resistance,its velocity $v$ varies with the height $h$ above the ground as:

The displacement of a particle moving in a straight line depends on time as $x = \alpha t^3 + \beta t^2 + \gamma t + \delta$. The ratio of initial acceleration to its initial velocity depends on:

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