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JEE Advance | PYQ

A particle of mass \( m \) is moving in the \( xy \)-plane such that its velocity at a point \( (x, y) \) is given as \( \vec{v} = \alpha(y\hat{x} + 2x\hat{y}) \), where \( \alpha \) is a non-zero constant. What is the force \( \vec{F} \) acting on the particle?

A projectile is thrown from a point \( O \) on the ground at an angle \( 45^\circ \) from the vertical and with a speed \( 5\sqrt{2} \, \text{m/s} \). The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground, \( 0.5 \, \text{s} \) after the splitting. The other part, \( t \) seconds after the splitting, falls to the ground at a distance \( x \) meters from the point \( O \). The acceleration due to gravity \( g = 10 \, \text{m/s}^2 \).

The value of \( t \) is _____

A projectile of mass \( 200 \, \text{g} \) is launched in a viscous medium at an angle \( 60^\circ \) with the horizontal, with an initial velocity of \( 270 \, \text{m/s} \). It experiences a viscous drag force \( \vec{F} = -c \vec{v} \) where the drag coefficient \( c = 0.1 \, \text{kg/s} \) and \( \vec{v} \) is the instantaneous velocity of the projectile. The projectile hits a vertical wall after \( 2 \, \text{s} \). Taking \( e = 2.7 \), the horizontal distance of the wall from the point of projection (in \( \text{m} \)) is ____.

A projectile is thrown from a point \( O \) on the ground at an angle \( 45^\circ \) from the vertical and with a speed \( 5\sqrt{2} \, \text{m/s} \). The projectile at the highest point of its trajectory splits into two equal parts. One part falls vertically down to the ground, \( 0.5 \, \text{s} \) after the splitting. The other part, \( t \) seconds after the splitting, falls to the ground at a distance \( x \) meters from the point \( O \). The acceleration due to gravity \( g = 10 \, \text{m/s}^2 \).

The value of \( x \) is _____

A projectile is fired from horizontal ground with speed \( v \) and projection angle \( \theta \). When the acceleration due to gravity is \( g \), the range of the projectile is \( d \). If at the highest point in its trajectory, the projectile enters a different region where the effective acceleration due to gravity is \( g' = \dfrac{g}{0.81} \) then the new range is \( d' = n d \). The value of \( n \) is ________.