### JEE Mains Previous Years Questions with Solutions

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1

### AIEEE 2005

The block of mass $M$ moving on the frictionless horizontal surface collides with the spring of spring constant $k$ and compresses it by length $L.$ The maximum momentum of the block after collision is
A
${{k{L^2}} \over {2M}}$
B
$\sqrt {Mk} \,\,L$
C
${{M{L^2}} \over k}$
D
Zero

## Explanation

Elastic energy stored in the spring = ${1 \over 2}k{L^2}$

And kinetic energy of the block = ${1 \over 2}M{v^2}$

$\therefore$ ${1 \over 2}M{v^2} = {1 \over 2}k{L^2}$

$\Rightarrow v = \sqrt {{k \over M}} .L$

$\therefore$ Momentum $= M \times v = M \times \sqrt {{k \over M}} .L = \sqrt {kM} .L$
2

### AIEEE 2005

A body of mass $m$ is accelerated uniformly from rest to a speed $v$ in a time $T.$ The instantaneous power delivered to the body as a function of time is given by
A
${{m{v^2}} \over {{T^2}}}.{t^2}$
B
${{m{v^2}} \over {{T^2}}}.t$
C
${1 \over 2}{{m{v^2}} \over {{T^2}}}.{t^2}$
D
${1 \over 2}{{m{v^2}} \over {{T^2}}}.t$

## Explanation

$u = 0;v = u + aT;v = aT$

Instantaneous power $= F \times v = m.\,a.\,at = m.{a^2}.t$

$\therefore$ Instantaneous power $= {{m{v^2}t} \over {{T^2}}}$
3

### AIEEE 2005

A spherical ball of mass $20$ $kg$ is stationary at the top of a hill of height $100$ $m$. It rolls down a smooth surface to the ground, then climbs up another hill of height $30$ $m$ and finally rolls down to a horizontal base at a height of $20$ $m$ above the ground. The velocity attained by the ball is
A
$20$ $m/s$
B
$40$ $m/s$
C
$10\sqrt {30} \,\,\,m/s$
D
$10\,\,m/s$

## Explanation

Loss in potential energy $=$ gain in kinetic energy

$m \times g \times 80 = {1 \over 2}m{v^2}$

$\Rightarrow$ $10 \times 80 = {1 \over 2}{v^2}$

$\Rightarrow$${v^2} = 1600$ or $v = 40\,m/s$
4

### AIEEE 2005

The upper half of an inclined plane with inclination $\phi$ is perfectly smooth while the lower half is rough. A body starting from rest at the top will again come to rest at the bottom if the coefficient of friction for the lower half is given by
A
$2\,\cos \,\,\phi$
B
$2\,sin\,\,\phi$
C
$\,\tan \,\,\phi$
D
$2\,\tan \,\,\phi$

## Explanation

Let the length of the inclined plane is = $l$. So only ${l \over 2}$ part will have friction.

According to work-energy theorem, $W = \Delta k = 0$ (Since initial and final speeds are zero)

$\therefore$ Work done by friction + Work done by gravity $=0$

i.e., $- \left( {\mu \,mg\,\cos \,\phi } \right){\ell \over 2} + mg\ell \,\sin \,\phi = 0$

or ${\mu \over 2}\cos \,\phi = \sin \phi$

or $\mu = 2\,\tan \,\phi$