Two particles of equal masses are revolving in circular paths of radii $r_1$ and $r_2$ respectively with the same speed. The ratio of their centripetal forces is

Two projectiles $A$ and $B$ are thrown with the same speed but at angles of $40^{\circ}$ and $50^{\circ}$ with the horizontal,respectively. Then:

When a body moves with a constant speed along a circle,

State whether the following statements are true or false:
$(a)$ If the angular velocity is constant on a circular path,the linear velocity is also constant.
$(b)$ In projectile motion,the velocity vector of the projectile is always perpendicular to the acceleration vector.
$(c)$ For the maximum horizontal range $R$ of a projectile,the maximum height attained is $\frac{R}{4}$.
$(d)$ If $|\vec{A} \times \vec{B}| = AB$,then the angle between $\vec{A}$ and $\vec{B}$ is zero.

$A$ particle moves over a $xy$ plane with a constant acceleration $\vec{a} = (4.0 \, m \, s^{-2}) \hat{i} + (4.0 \, m \, s^{-2}) \hat{j}$. At time $t = 0$, the velocity is $\vec{v}_0 = (4.0 \, m \, s^{-1}) \hat{i}$. The speed of the particle when it is displaced by $6.0 \, m$ parallel to the $x$-axis is,

$A$ projectile of mass $m$ is fired with velocity $v$ from the point $P$ at an angle $45^{\circ}$ with the horizontal. The magnitude of change in momentum,when it passes through the point $Q$ on the same horizontal line on which $P$ lies,is :