Distance x covered by a particle in one dimensional motion varies with time t as mathrm{x}^{2}=mathr

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2.Motion in Straight Line

hard

The distance $x$ covered by a particle in one dimensional motion varies with time $t$ as $\mathrm{x}^{2}=\mathrm{at}^{2}+2 \mathrm{bt}+\mathrm{c.}$ If the acceleration of the particle depends on $\mathrm{x}$ as $\mathrm{x}^{-\mathrm{n}},$ where $\mathrm{n}$ is an integer, the value of $\mathrm{n}$ is

$9$

$6$

$4$

$3$

(JEE MAIN-2020)

Solution

$\mathrm{x}=\sqrt{\mathrm{at}^{2}+2 \mathrm{bt}+\mathrm{c}}$

Differentiating w.r.t. time

$\frac{d x}{d t}=v=\frac{1}{2 \sqrt{a t^{2}+2 b t+c}} \times(2 a t+2 b)$

$\Rightarrow \mathrm{v}=\frac{\mathrm{at}+\mathrm{b}}{\mathrm{x}}$

$\Rightarrow \mathrm{vx}=\mathrm{at}+\mathrm{b}$

Differentiating w.r.t. $\mathrm{x}$

$\Rightarrow \frac{d v}{d x} \times x+v=a \times \frac{d t}{d x}$

Multiply both side by v

$\Rightarrow\left(v \frac{d v}{d x}\right) x+v^{2}=a$

$\Rightarrow \mathrm{a}^{\prime} \mathrm{x}=\mathrm{a}-\mathrm{v}^{2}$ [Here $a'$ is acceleration]

$\Rightarrow \mathrm{a}^{\prime} \mathrm{x}=\mathrm{a}-\left(\frac{\mathrm{at}+\mathrm{b}}{\mathrm{x}}\right)^{2}$

$\Rightarrow \mathrm{a}^{\prime} \mathrm{x}=\frac{\mathrm{ax}^{2}-(\mathrm{at}+\mathrm{b})^{2}}{\mathrm{x}^{2}}$

$\Rightarrow \mathrm{a}^{\prime} \mathrm{x}=\frac{\mathrm{a}\left(\mathrm{at}^{2}+2 \mathrm{bt}+\mathrm{c}\right)-(\mathrm{at}+\mathrm{b})^{2}}{\mathrm{x}^{2}}$

$\Rightarrow \mathrm{a}^{\prime} \mathrm{x}=\frac{\mathrm{ac}-\mathrm{b}^{2}}{\mathrm{x}^{2}}$

$\Rightarrow \mathrm{a}^{\prime}=\frac{\mathrm{ac}-\mathrm{b}^{2}}{\mathrm{x}^{3}}$

$\therefore a^{\prime} \propto \frac{1}{x^{3}} \quad \therefore n=3$

Standard 11

Physics

The acceleration $'a'$ in $m/{s^2}$ of a particle is given by $a = 3{t^2} + 2t + 2$ where $t$ is the time. If the particle starts out with a velocity $u = 2\,m/s$ at $t = 0$, then the velocity at the end of $2$ second is…….$m/s$

medium

A particle executes the motion described by $x(t) = x_0 (1 – e^{-\gamma t} )$ ; જ્યાં $t\, \geqslant \,0\,,\,{x_0}\, > \,0$.

$(a)$ Where does the particle start and with what velocity ?

$(b)$ Find maximum and minimum values of $x(t),\, v(t)$ $a(t)$. Show that $x(t)$ and $a(t)$ increase with time and $v(t)$ decreases with time.

hard

A particle moves along a straight line such that its displacement at any time $t$ is given by $s = (t^3 -6t^2 + 3t + 4)\, m$. …… $m/s$ is the velocity of the particle when its acceleration is zero …………….. $\mathrm{m/s}$

medium

If the velocity of a body related to displacement ${x}$ is given by $v=\sqrt{5000+24 {x}} \;{m} / {s}$, then the acceleration of the body is $\ldots \ldots {m} / {s}^{2}$

hard

(JEE MAIN-2021)

Match the following columns.

Colum $I$ Colum $II$

$(A)$ $\frac{dv}{dt}$ $(p)$ Acceleration

$(B)$ $\frac{d|v|}{dt}$ $(q)$ Magnitude of acceleration

$(C)$ $\frac{dr}{dt}$ $(r)$ Velocity

$(D)$ $\left|\frac{d r }{d t}\right|$ $(s)$ Magnitude of velocity

easy

Colum $I$	Colum $II$
$(A)$ $\frac{dv}{dt}$	$(p)$ Acceleration
$(B)$ $\frac{d\|v\|}{dt}$	$(q)$ Magnitude of acceleration
$(C)$ $\frac{dr}{dt}$	$(r)$ Velocity
$(D)$ $\left\|\frac{d r }{d t}\right\|$	$(s)$ Magnitude of velocity

Solution

Similar Questions

The acceleration $'a'$ in $m/{s^2}$ of a particle is given by $a = 3{t^2} + 2t + 2$ where $t$ is the time. If the particle starts out with a velocity $u = 2\,m/s$ at $t = 0$, then the velocity at the end of $2$ second is…….$m/s$

A particle moves along a straight line such that its displacement at any time $t$ is given by $s = (t^3 -6t^2 + 3t + 4)\, m$. …… $m/s$ is the velocity of the particle when its acceleration is zero …………….. $\mathrm{m/s}$

If the velocity of a body related to displacement ${x}$ is given by $v=\sqrt{5000+24 {x}} \;{m} / {s}$, then the acceleration of the body is $\ldots \ldots {m} / {s}^{2}$

Match the following columns. Colum $I$ Colum $II$ $(A)$ $\frac{dv}{dt}$ $(p)$ Acceleration $(B)$ $\frac{d|v|}{dt}$ $(q)$ Magnitude of acceleration $(C)$ $\frac{dr}{dt}$ $(r)$ Velocity $(D)$ $\left|\frac{d r }{d t}\right|$ $(s)$ Magnitude of velocity

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Match the following columns.

Colum $I$ Colum $II$

$(A)$ $\frac{dv}{dt}$ $(p)$ Acceleration

$(B)$ $\frac{d|v|}{dt}$ $(q)$ Magnitude of acceleration

$(C)$ $\frac{dr}{dt}$ $(r)$ Velocity

$(D)$ $\left|\frac{d r }{d t}\right|$ $(s)$ Magnitude of velocity