$A$ projectile is fired at a speed of $100 \, m/s$ at an angle of $37^\circ$ 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 ............... $m$.

Let $\overline{a}=2 \hat{i}+\hat{j}-2 \hat{k}$ and $\overline{b}=\hat{i}+\hat{j}$. Let $\overline{c}$ be a vector such that $|\overline{c}-\overline{a}|=3$,$|(\overline{a} \times \overline{b}) \times \overline{c}|=3$,and the angle between $\overline{c}$ and $\overline{a} \times \overline{b}$ is $30^{\circ}$. Then $\overline{a} \cdot \overline{c}$ is equal to

On closing an open organ pipe from one end,it is noticed that the frequency of the third harmonic is $50 ~Hz$ more than the fundamental frequency of vibration in the open organ pipe. The fundamental frequency of the open organ pipe is (in $~Hz$)

Why is a solution of potassium hydroxide used to absorb carbon dioxide evolved during the estimation of carbon present in an organic compound?

$A$ chloro compound of Vanadium has a spin-only magnetic moment of $1.73 \, BM$. This Vanadium chloride has the formula:

$A$ metallic ring of radius $r$ and cross-sectional area $A$ is fitted into a wooden circular disc of radius $R$ $(R > r)$. If the Young's modulus of the material of the ring is $Y$,the force with which the metal ring expands is