Q4. A shell fired on level ground would land 200 m away. At the top of its path it breaks into two pieces with masses in the ratio 1 : 3. The lighter piece retraces its path back to the gun. Where does the heavier piece land? Ignore air resistance.
At the top the shell moves horizontally at some speed u. The light piece (mass m) goes back at −u, so momentum conservation for the 4m shell gives 4mu = m(−u) + 3mv, and v = 5u/3, also horizontal. Both pieces start with no vertical velocity, so they land together, and their centre of mass lands at 200 m. The light piece lands at 0, so
(1 × 0 + 3x)/4 = 200, which gives x ≈ 267 m (exactly 800/3 m).
Q5. A man of mass 50 kg stands at one end of a plank 10 m long and of mass 200 kg, lying on smooth ice. He walks to the other end. How far does the plank move, and how far does the man move relative to the ice?
The centre of mass stays at rest. With the plank moving xp and the man xm relative to the ice: 50xm + 200xp = 0 and xm − xp = 10. So 250xp = −500, xp = −2 m and xm = 8 m. The plank slides 2 m back and the man moves 8 m forward.
Q6. Two balls are thrown at the same instant from different points, in different directions and with different speeds. What is the acceleration of their centre of mass while both are in the air? Ignore air resistance.
The only external forces are their weights, m1g and m2g, so (m1 + m2)A = (m1 + m2)g, and A = g, vertically downward (about 9.8 m s), whatever the paths. The centre of mass moves like a single projectile.