A uranium-$238$ nucleus, initially at rest, emits an alpha particle with a speed of $1.4\times {10}^7 \mathrm{m} {\mathrm{s}}^{-1}$. Calculate the recoil speed of the residual nucleus thorium-$234$. Assume that the mass of a nucleus is proportional to the mass number.

A man of mass $50 \mathrm{kg}$ starts moving on the Earth and acquires a speed of $1.8 \mathrm{m} {\mathrm{s}}^{-1}$. With what speed does the Earth recoil? Mass of the Earth $=6\times {10}^{24} \mathrm{kg}$.

A neutron initially at rest, decays into a proton, an electron and an antineutrino. The ejected electron has a momentum of $1.4\times {10}^{-26} \mathrm{kg}-\mathrm{m} {\mathrm{s}}^{-1}$ and the antineutrino, $6.4\times {10}^{-27} \mathrm{kg}-\mathrm{m} {\mathrm{s}}^{-1}$. Find the recoil speed of the proton,
(a) if the electron and the antineutrino are ejected along the same direction and
(b) if they are ejected along perpendicular directions. Mass of the proton $=1.67\times {10}^{-27} \mathrm{kg}$.

A man of mass $M$ having a bag of mass $m$ slips from the roof of a tall building of height $H$ and starts falling vertically (figure). When at a height $h$ from the ground, he notices that the ground below him is pretty hard, but there is a pond at a horizontal distance $x$ from the line of fall. In order to save himself, he throws the bag horizontally (with respect to himself) in the direction opposite to the pond. Calculate the minimum horizontal velocity imparted to the bag so that the man lands in the water. If the man just succeeds to avoid the hard ground, where will the bag land?

A ball of mass $50 \mathrm{g}$ moving at a speed of $2.0 \mathrm{m} {\mathrm{s}}^{-1}$ strikes a plane surface at an angle of incidence $45°.$ The ball is reflected by the plane at equal angle of reflection with the same speed. Calculate
(a) the magnitude of the change in momentum of the ball;

(b) the change in the magnitude of the momentum of the ball.

A block at rest explodes into three equal parts. Two parts start moving along $X$ and $Y$ axes, respectively, with equal speeds of $10 \mathrm{m} {\mathrm{s}}^{-1}$. Find the initial velocity of the third part.

Two fat astronauts each of mass $120 \mathrm{kg}$ are travelling in a closed spaceship moving at a speed of $15 \mathrm{km} {\sec }^{-1}$ in the outer space far removed from all other material objects. The total mass of the spaceship and its contents including the astronauts is $660 \mathrm{kg}$. If the astronauts do slimming exercise and thereby reduce their masses to $90 \mathrm{kg}$ each, with what velocity will the spaceship move?