What is the value of linear velocity,if $\vec \omega = 3\hat i - 4\hat j + \hat k$ and $\vec r = 5\hat i - 6\hat j + 6\hat k$?

Two objects of equal mass are projected with the same velocity $v$ at the same angle $\theta$ towards each other from a distance $R$. What is the momentum of the system at the moment of collision?

At a given instant of time,two particles have position vectors $4 \hat{i} + 4 \hat{j} + 57 \hat{k} \ m$ and $2 \hat{i} + 2 \hat{j} + 5 \hat{k} \ m$ respectively. If the velocity of the first particle is $0.4 \hat{i} \ ms^{-1}$,what is the velocity of the second particle in $ms^{-1}$ if they collide after $10 \ s$?

$A$ merry-go-round starts from rest and accelerates at $0.4 \ rad \ s^{-2}$ for the first $5 \ s$. It then rotates at this constant angular velocity for $30 \ s$ and finally comes to rest with the same magnitude of angular deceleration. What is the total linear distance traveled by a child sitting $3 \ m$ from the center of the merry-go-round?

$A$ projectile of mass $m$ is projected from the ground with a speed of $50 \, m/s$ at an angle of $53^{\circ}$ with the horizontal. It breaks up into two equal parts at the highest point of the trajectory. One particle comes to rest immediately after the explosion. The ratio of the radii of curvatures of the moving particle just before and just after the explosion is:

$A$ projectile is thrown with velocity $v$ making an angle $\theta$ with the vertical. It reaches a maximum height $H$. The time of flight is: