$A$ projectile is fired at a speed of $100 \, m/s$ at an angle of $37^o$ above the horizontal. At the highest point,the projectile breaks into two parts of mass ratio $1:3$,the smaller coming to rest. Then the distance of the heavier part from the launching point is ........... $m$.

Find the speed of the intersection point $O$ of the two wires if the wires start moving perpendicular to themselves with speed $v$ as shown in the figure.

The position coordinates of a projectile projected from the ground on a certain planet (with no atmosphere) are given by $y = (4t - 2t^2) \text{ m}$ and $x = (3t) \text{ m}$,where $t$ is in seconds and the point of projection is taken as the origin. The angle of projection of the projectile with the vertical is ......... (in $^{\circ}$)

$A$ projectile is launched at an angle $\alpha$ with the horizontal with a velocity $20 \; m/s$. After $10 \; s$,its inclination with the horizontal is $\beta$. The value of $\tan \beta$ will be: $(g = 10 \; m/s^2)$

$A$ boy is moving with a constant speed $v$ on a small trolley towards a distant circle as shown in the figure. $A$ point mass is moving on the circle with a constant speed $v$. What is the frequency of change in the magnitude of the relative velocity of the point mass,as observed by the boy?

Consider a circle of radius $R = 42\ cm$. An insect crawls with a uniform speed of $v = 1.3\ cm/s$ along the chord $AB$, then along the circular arc $BCD$, and finally along the chord $DA$ to return to point $A$. The time taken by the insect to crawl from $A$ back to $A$ is closest to ......... $\sec$.