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A person travels \( x \) distance with velocity \( v_1 \) and then \( x \) distance with velocity \( v_2 \) in the same direction. The average velocity of the person is \( v \), then the relation between \( v \), \( v_1 \) and \( v_2 \) will be:

Two bodies are projected from ground with same speeds \( 40 \, \text{m s}^{-1} \) at two different angles with respect to horizontal. The bodies were found to have same range. If one of the body was projected at an angle of \( 60^\circ \), with horizontal then sum of the maximum heights, attained by the two projectiles, is ______ m.
(Given \( g = 10 \, \text{m s}^{-2} \))

For a body projected at an angle with the horizontal from the ground, choose the correct statement.

As shown in the figure, a particle is moving with constant speed \( \pi \, \text{m/s} \). Considering its motion from \( A \) to \( B \), the magnitude of the average velocity is:

For a train engine moving with speed of \( 20 \, \text{m s}^{-1} \), the driver must apply brakes at a distance of \( 500 \, \text{m} \) before the station for the train to come to rest at the station. If the brakes were applied at half of this distance, the train engine would cross the station with speed \( \sqrt{x} \, \text{m s}^{-1} \). The value of \( x \) is _______. (Assume same retardation is produced by brakes)

Two trains '\( A \)' and '\( B \)' of length '\( l \) 'and '\( 4l \)' are travelling into a tunnel of length '\( L \)' in parallel tracks from opposite directions with velocities \( 108 \, \text{km/h} \) and \( 72 \, \text{km/h} \), respectively. If train' \( A \)' take \( 35 \, \text{s} \) less time than train '\( B \)' to cross the tunnel then, length '\( L \)' of tunnel is: (Given \( L = 60 \, l \))

The initial speed of a projectile fired from ground is \( u \). At the highest point during its motion, the speed of projectile is \( \frac{\sqrt{3}}{2}u \). The time of flight of the projectile is:

The maximum vertical height to which a man can throw a ball is \( 136 \, \text{m} \). The maximum horizontal distance upto which he can throw the same ball is:

A particle is moving with constant speed in a circular path. When the particle turns by an angle \( 90^\circ \), the ratio of instantaneous velocity to its average velocity is \( \pi : x\sqrt{2} \). The value of \( x \) will be:

The trajectory of projectile, projected from the ground is given by \( y = x - \dfrac{x^2}{20} \). Where \( x \) and \( y \) are measured meter. The maximum height attained by the projectile will be:

Two particles A and B are thrown with initial velocities of \( 40 \, \text{m/s} \) and \( 60 \, \text{m/s} \) at angles \( 30^\circ \) and \( 60^\circ \) with the horizontal respectively. The ratio of their ranges respectively is:
\( (g = 10 \, \text{m/s}^2) \)

Two objects are projected with same velocity ‘\( u \)’ however at different angles \( \alpha \) and \( \beta \) with the horizontal. If \( \alpha + \beta = 90^\circ \), the ratio of horizontal range of the first object to the 2nd object will be:

A child stands on the edge of the cliff \( 10 \, \text{m} \) above the ground and throws a stone horizontally with an initial speed of \( 5 \, \text{m}\,\text{s}^{-1} \). Neglecting the air resistance, the speed with which the stone hits the ground will be ________ \( \text{m}\,\text{s}^{-1} \). (given, \( g = 10 \, \text{m}\,\text{s}^{-2} \))

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : When a body is projected at an angle \( 45^\circ \), it's range is maximum.
Reason R : For maximum range, the value of \( \sin 2\theta \) should be equal to one.
In the light of the above statements, choose the correct answer from the options given below:

The range of the projectile projected at an angle of \( 15^\circ \) with horizontal is \( 50 \, \text{m} \). If the projectile is projected with same velocity at an angle of \( 45^\circ \) with horizontal, then its range will be:

A body is moving with constant speed, in a circle of radius \( 10 \, \text{m} \). The body completes one revolution in \( 4 \, \text{s} \). At the end of \( 3^{\text{rd}} \) second, the displacement of body (in \( \text{m} \)) from its starting point is:

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JEE Main 2023 31st Jan Morning Shift

The initial speed of a projectile fired from ground is \( u \). At the highest point during its motion, the speed of projectile is \( \frac{\sqrt{3}}{2}u \). The time of flight of the projectile is: