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A nucleus disintegrates into two nuclear parts, in such a way that ratio of their nuclear sizes is \( 1 : 2^{1/3} \). Their respective speed have a ratio of \( n : 1 \). The value of \( n \) is ______.

If the maximum load carried by an elevator is \( 1400 \, \text{kg} \) (\( 600 \, \text{kg} \) – passengers + \( 800 \, \text{kg} \) – elevator), which is moving up with a uniform speed of \( 3 \, \text{m s}^{-1} \) and the frictional force acting on it is \( 2000 \, \text{N} \), then the maximum power used by the motor is ______ \( \text{kW} \) (\( g = 10 \, \text{m/s}^2 \)).

A coin placed on a rotating table just slips when it is placed at a distance of \( 1 \, \text{cm} \) from the center. If the angular velocity of the table in halved, it will just slip when placed at a distance of _______ from the centre.

A block is fastened to a horizontal spring. The block is pulled to a distance \( x = 10 \,\text{cm} \) from its equilibrium position (at \( x = 0 \)) on a frictionless surface from rest. The energy of the block at \( x = 5 \,\text{cm} \) is \( 0.25 \,\text{J} \). The spring constant of the spring is _______ \( \text{Nm}^{-1} \).

A block of mass \( 5 \, \text{kg} \) starting from rest pulled up on a smooth incline plane making an angle of \( 30^\circ \) with horizontal with an affective acceleration of \( 1 \, \text{m s}^{-2} \). The power delivered by the pulling force at \( t = 10 \, \text{s} \) from the start is ________ \( \text{W} \) [use \( g = 10 \, \text{m s}^{-2} \)] (calculate the nearest integer value)

A passenger sitting in a train A moving at \( 90 \, \text{km/h} \) observes another train B moving in the opposite direction for \( 8 \, \text{s} \). If the velocity of the train B is \( 54 \, \text{km/h} \), then length of train B is:

A body of mass \( 2 \, \text{kg} \) is initially at rest. It starts moving unidirectionally under the influence of a source of constant power \( P \). Its displacement in \( 4 \, \text{s} \) is \( \dfrac{1}{3} \alpha^{2} \sqrt{P} \, \text{m} \). The value of \( \alpha \) will be _____.

Two projectiles are projected at \( 30^\circ \) and \( 60^\circ \) with the horizontal with the same speed. The ratio of the maximum height attained by the two projectiles respectively is:

As shown in the figure a block of mass \( 10 \, \text{kg} \) lying on a horizontal surface is pulled by a force \( F \) acting at an angle \( 30^\circ \), with horizontal. For \( \mu_s = 0.25 \), the block will just start to move for the value of \( F \):
[Given \( g = 10 \, \text{m s}^{-2} \)]

Identify the correct statements from the following:
(A) Work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket is negative
(B) Work done by gravitational force in lifting a bucket out of a well by a rope tied to the bucket is negative
(C) Work done by friction on a body sliding down an inclined plane is positive
(D) Work done by an applied force on a body moving on a rough horizontal plane

The figure represents the momentum time \( (p - t) \) curve for a particle moving along an axis under the influence of the force. Identify the regions on the graph where the magnitude of the force is maximum and minimum respectively? If \( (t_{3} - t_{2}) < t_{1} \).

Figure (a), (b), (c) and (d) show variation of force with time:

The impulse is highest in figure:

A stone tied to \( 180 \, \text{cm} \) long string at its end is making \( 28 \) revolutions in horizontal circle in every minute. The magnitude of acceleration of stone is \( \dfrac{1936}{x} \, \text{m s}^{-2} \). The value of \( x \) ___. \( \left( \text{Take } \pi = \dfrac{22}{7} \right) \)

Figure below shows a liquid being pushed out of the tube by a piston having area of cross section \( 2.0 \, \text{cm}^2 \). The area of cross section at the outlet is \( 10 \, \text{mm}^2 \). If the piston is pushed at a speed of \( 4 \, \text{cm s}^{-1} \), the speed of outgoing fluid is ______ \( \text{cm s}^{-1} \).

A block of \( \sqrt{3} \, \text{kg} \) is attached to a string whose other end is attached to the wall. An unknown force \( F \) is applied so that the string makes an angle of \( 30^\circ \) with the wall. The tension \( T \) is: ( Given \( g = 10 \, \text{m s}^{-2} \) )

A projectile fired at \( 30^\circ \) to the ground is observed to be at same height at time \( 3 \, \text{s} \) and \( 5 \, \text{s} \) after projection, during its flight. The speed of projection of the projectile is _________ \( \text{m s}^{-1} \).
(Given \( g = 10 \, \text{m s}^{-2} \))

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JEE Main 2023 10th April Evening Shift

Two projectiles are projected at \( 30^\circ \) and \( 60^\circ \) with the horizontal with the same speed. The ratio of the maximum height attained by the two projectiles respectively is: