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A ball is projected with kinetic energy \( E \), at an angle of \( 60^\circ \) to the horizontal. The kinetic energy of this ball at the highest point of its flight will become:

In Bohr’s atomic model of hydrogen, let \( K \), \( P \) and \( E \) are the kinetic energy, potential energy and total energy of the electron respectively. Choose the correct option when the electron undergoes transitions to a higher level:

A ball is spun with angular acceleration \( \alpha = 6t^2 - 2t \) where \( t \) is in second and \( \alpha \) is in \( \text{rad s}^{-2} \). At \( t = 0 \), the ball has angular velocity of \( 10 \, \text{rad s}^{-1} \) and angular position of \( 4 \, \text{rad} \). The most appropriate expression for the angular position of the ball is:

A body of mass \( 10 \, \text{kg} \) is projected at an angle of \( 45^\circ \) with the horizontal. The trajectory of the body is observed to pass through a point \( (20, 10) \). If \( T \) is the time of flight, then its momentum vector, at time \( t = \dfrac{T}{\sqrt{2}} \), is __________ [Take \( g = 10 \, \text{m}/\text{s}^{2} \)]

A fly wheel is accelerated uniformly from rest and rotates through \( 5 \,\text{rad} \) in the first second. The angle rotated by the fly wheel in the next second, will be:

A body is projected from the ground at an angle of \( 45^\circ \) with the horizontal. Its velocity after \( 2 \, \text{s} \) is \( 20 \, \text{ms}^{-1} \). The maximum height reached by the body during its motion is _______ \( \text{m} \). (use \( g = 10 \, \text{ms}^{-2} \) )

A projectile is launched at an angle ‘\( \alpha \)’ with the horizontal with a velocity \( 20 \, \text{m s}^{-1} \). After \( 10 \, \text{s} \), its inclination with horizontal is ‘\( \beta \)’. The value of \( \tan \beta \) will be: \( ( \, g = 10 \, \text{m /s}^{2} \, ) \)

A projectile is projected with velocity of \( 25 \, \text{m/s} \) at an angle \( \theta \) with the horizontal. After \( t \) seconds its inclination with horizontal becomes zero. If \( R \) represents horizontal range of the projectile, the value of \( \theta \) will be: [use \( g = 10 \, \text{m/s}^{2} \)]

A person can throw a ball up to a maximum range of \( 100 \, \text{m} \). How high above the ground he can throw the same ball?

Motion of a particle in \( x-y \) plane is described by a set of following equations \( x=4\sin\left(\frac{\pi}{2}-\omega t\right)\, \text{m} \) and \( y=4\sin(\omega t)\, \text{m} \). The path of the particle will be:

If the initial velocity in horizontal direction of a projectile is unit vector \( \hat{i} \) and the equation of trajectory is \( y = 5x(1 - x) \). The y component vector of the initial velocity is ________ \( \hat{j} \).
(Take \( g = 10 \, \text{m}/\text{s}^2 \))

A force on an object of mass \( 100 \, \text{g} \) is \( (10\hat{i} + 5\hat{j}) \, \text{N} \). The position of that object at \( t = 2 \, \text{s} \) is \( (a\hat{i} + b\hat{j}) \, \text{m} \) after starting from rest. The value of \( \frac{a}{b} \) will be __________________.

An object projected in the air with initial velocity \( u \) at an angle \( \theta \). The projectile motion is such that the horizontal range \( R \), is maximum. Another object is projected in the air with a horizontal range half of the range of first object. The initial velocity remains same in both the case. The value of the angle of projection, at which the second object is projected, will be _________ degree.

Two projectiles are thrown with same initial velocity making an angle of \( 45^\circ \) and \( 30^\circ \) with the horizontal respectively. The ratio of their respective ranges will be:

A ball is projected from the ground with a speed \( 15 \, \text{ms}^{-1} \) at an angle \( \theta \) with horizontal so that its range and maximum height are equal, then ‘\( \tan \theta \)’ will be equal to:

A ball of mass \( m \) is thrown vertically upward. Another ball of mass \( 2m \) is thrown at an angle \( \theta \) with the vertical. Both the balls stay in air for the same period of time. The ratio of the heights attained by the two balls respectively is \( {1}/{x} \). The value of \( x \) is _______.

Two projectiles thrown at \( 30^\circ \) and \( 45^\circ \) with the horizontal respectively, reach the maximum height in same time. The ratio of their initial velocities is:

A fighter jet is flying horizontally at a certain altitude with a speed of \( 200 \, \text{ms}^{-1} \). When it passes directly overhead an anti-aircraft gun, a bullet is fired from the gun, at an angle \( \theta \) with the horizontal, to hit the jet. If the bullet speed is \( 400 \, \text{m/s} \), the value of \( \theta \) will be ____________\( ^\circ \).

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JEE Main 2022 26th July Evening Shift

Two projectiles are thrown with same initial velocity making an angle of \( 45^\circ \) and \( 30^\circ \) with the horizontal respectively. The ratio of their respective ranges will be: