$A$ bomb of mass $1 \,kg$ initially at rest,explodes and breaks into three fragments of masses in the ratio $1: 1: 3$. The two pieces of equal mass fly off perpendicular to each other with a speed $15 \,m / s$ each. The speed of the heavier fragment is ........... $m / s$.

$A$ cannon ball is fired with a velocity $200 \, m/s$ at an angle of $60^{\circ}$ with the horizontal. At the highest point of its flight,it explodes into $3$ equal fragments. One fragment goes vertically upwards with a velocity $100 \, m/s$,and the second one falls vertically downwards with a velocity $100 \, m/s$. The third fragment will be moving with a velocity:

An object of mass $2m$ is projected with a speed of $100 \ m/s$ at an angle $\theta = \sin^{-1}(3/5)$ to the horizontal. At the highest point, the object breaks into two pieces of same mass $m$ and the first one comes to rest. The distance between the point of projection and the point of landing of the second piece (in metres) is $(g = 10 \ m/s^2)$.

$A$ buggy of mass $100\, kg$ is free to move on a frictionless horizontal track. Two men,each of mass $50\, kg$,are standing on the buggy,which is initially stationary. The men jump off the buggy with a velocity of $10\, m/s$ relative to the buggy. In one situation,the men jump one after the other. In another situation,the men jump simultaneously. What is the ratio of the recoil velocities of the buggy in the two cases?

$A$ bomb of mass $9 \,kg$ explodes into two pieces of masses $3 \,kg$ and $6 \,kg$. The velocity of mass $3 \,kg$ is $16 \,m/s$. The kinetic energy of mass $6 \,kg$ in joule is

$A$ body of mass $M$ at rest explodes into three pieces,in the ratio of masses $1: 1: 2$. Two smaller pieces fly off perpendicular to each other with velocities of $30 \, m/s$ and $40 \, m/s$ respectively. The velocity of the third piece will be ............... $\, m/s$.