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The co-ordinates of a particle moving in x-y plane are given by: \( x = 2 + 4t \) , \( y = 3t + 8t^{2} \). The motion of the particle is:

A given object takes \( n \) times the time to slide down \( 45^\circ \) rough inclined plane as it takes the time to slide down an identical perfectly smooth \( 45^\circ \) inclined plane. The coefficient of kinetic friction between the object and the surface of inclined plane is:

The angle of projection for a projectile to have same horizontal range and maximum height is:

A particle is moving in a circle of radius \( 50 \, \text{cm} \) in such a way that at any instant the normal and tangential components of it's acceleration are equal. If its speed at \( t = 0 \) is \( 4 \, \text{m/s} \), the time taken to complete the first revolution will be \( \dfrac{1}{\alpha} \big[ 1 - e^{-2\pi} \big] \, \text{s} \). where \( \alpha = \) __________.

Projectiles A and B are thrown at angles of \( 45^\circ \) and \( 60^\circ \) with vertical respectively from top of a \( 400 \, \text{m} \) high tower. If their range and times of flight are same, the ratio of their speeds of projection \( v_A : v_B \) is:
[Take \( g = 10 \, \text{m s}^{-2} \) ]

The maximum height reached by a projectile is \( 64 \, \text{m} \). If the initial velocity is halved, the new maximum height of the projectile is _____ \( \text{m} \).

A particle starts from origin at $t = 0$ with a velocity $5 \hat{i}\text{ m/s}$ and moves in x-y plane under action of a force which produces a constant acceleration of $(3\hat{i} + 2\hat{j})\text{ m/s}^2$. If the x-coordinate of the particle at that instant is $84\text{ m}$, then the speed of the particle at this time is $\sqrt{\alpha}\text{ m/s}$. The value of $\alpha$ is ___________.

A particle moving in a circle of radius \( R \) with uniform speed takes time \( T \) to complete one revolution. If this particle is projected with the same speed at an angle \( \theta \) to the horizontal, the maximum height attained by it is equal to \( 4R \). The angle of projection \( \theta \) is then given by:

A body of mass \( M \) thrown horizontally with velocity \( v \) from the top of the tower of height \( H \) touches the ground at a distance of \( 100 \, \text{m} \) from the foot of the tower. A body of mass \( 2M \) thrown at a velocity \(\displaystyle \frac{v}{2} \) from the top of the tower of height \( 4H \) will touch the ground at a distance of __________ \( \text{m} \).

A man carrying a monkey on his shoulder does cycling smoothly on a circular track of radius \( 9 \, \text{m} \) and completes \( 120 \) resolutions in \( 3 \) minutes. The magnitude of centripetal acceleration of monkey is (in \( {m} \, /{s}^{-2} \)):

A body of mass \(' m '\) is projected with a speed \( 'u' \) making an angle of \( 45^\circ \) with the ground. The angular momentum of the body about the point of projection, at the highest point is expressed as \( \dfrac{\sqrt{2}\, m u^{3}}{X g} \). The value of ‘\( X \)’ is ______________.

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JEE Main 2024 8th April Evening Shift

The angle of projection for a projectile to have same horizontal range and maximum height is: