The coordinates of a moving particle at any time $t$ are given by $x = a\, t^2$ and $y = b\, t^2$. The speed of the particle is

An insect trapped in a circular groove of radius $12 \; cm$ moves along the groove steadily and completes $7$ revolutions in $100 \; s$.
$(a)$ What is the angular speed,and the linear speed of the motion?
$(b)$ Is the acceleration vector a constant vector? What is its magnitude?

$A$ stone is projected vertically upwards with velocity $V$. Another stone of the same mass is projected at an angle of $60^{\circ}$ with the vertical with the same speed $V$. The ratio of their potential energies at the highest points of their journey is:

$A$ ball is thrown from the ground at an angle $\theta$ with the horizontal and with an initial speed $u_0$. For the resulting projectile motion,the magnitude of the average velocity of the ball up to the point when it hits the ground for the first time is $V_1$. After hitting the ground,the ball rebounds at the same angle $\theta$ but with a reduced speed of $u_0 / \alpha$. Its motion continues for a long time as shown in the figure. If the magnitude of the average velocity of the ball for the entire duration of motion is $0.8 V_1$,the value of $\alpha$ is:

$A$ projectile is thrown at an angle $\theta$ to the horizontal from $O$ and it hits the ground at $O'$. During the entire motion (select the wrong statement):

$A$ boy ties a stone of mass $100 \,g$ to the end of a $2 \,m$ long string and whirls it around in a horizontal plane. The string can withstand a maximum tension of $80 \,N$. If the maximum speed with which the stone can revolve is $\frac{K}{\pi} \,rev/min$,find the value of $K$. (Assume the string is massless and unstretchable)