$A$ particle of mass $m$ moves along a circular path of constant radius $r$. The centripetal acceleration of the particle $a_c$ varies with time $t$ as $a_c = k^2rt^2$,where $k$ is a constant. What is the power delivered to the particle by the force acting on it?

Shown here are the velocity and acceleration vectors for a man in several different types of motion. In which case is the man slowing down and turning to the right?

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

$A$ man wants to reach from $A$ to the opposite corner of the square $C$. The sides of the square are $100\, m$. $A$ central square of $50\, m \times 50\, m$ is filled with sand. Outside this square,he can walk at a speed of $1\, m/s$. In the central square,he can walk only at a speed of $v\, m/s$ $(v < 1)$. What is the smallest value of $v$ for which he can reach faster via a straight path through the sand than any path in the square outside the sand?

$A$ 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:

Three particles start simultaneously from a point on a horizontal smooth plane. The first particle moves with speed $v_1$ towards the east,the second particle moves towards the north with speed $v_2$,and the third one moves towards the north-east. The velocity of the third particle,such that the three particles always lie on a straight line,is