Q.A spherical ball of mass 0.5 kg is dropped from some height. On falling freely for 10 s, it explodes into two fragments of mass ratio 1:2. The lighter fragment continues to travel downwards with a speed of 60 m/s. Calculate the kinetic energy supplied during the explosion.

[!TLDR] Using momentum conservation with a 1:2 mass split and the lighter fragment's given speed, then the kinetic-energy difference before/after …

Total mass 0.5 kg splits into fragments of mass ratio 1:2, i.e. $m_1=\frac{1}{3}(0.5)=\frac{1}{6}$ kg (lighter) and $m_2=\frac{2}{3}(0.5)=\frac{1}{3}$ kg (heavier). Just before explosion, after falling freely for 10 s, $u=gt=10\times10=100$ m/s (downward), so the whole 0.5 kg ball has speed 100 m/s downward just before exploding. The lighter fragment continues downward at $v_1=60$ m/s (given). By conservation of momentum (taking downward as positive): $0.5(100)=\frac{1}{6}(60)+\frac{1}{3}v_2\Rightarrow 50=10+\frac{1}{3}v_2\Rightarrow \frac{1}{3}v_2=40\Rightarrow v_2=120$ m/s (downward). KE before explosion $=\frac{1}{2}(0.5)(100)^2=2500$ J. KE after explosion $=\frac{1}{2}\left(\frac{1}{6}\right)(60)^2+\frac{1}{2}\left(\frac{1}{3}\right)(120)^2=\frac{1}{2}\times\frac{1}{6}\times3600+\frac{1}{2}\times\frac{1}{3}\times14400=300+2400=2700$ J. KE supplied …

Find the pre-explosion speed from free fall (v=gt), use momentum conservation to find the unknown (heavier) fragment's speed given the lighter frag …

Mis-assigning which fragment gets which fraction of the total mass (1/3 vs 2/3) from the stated 1:2 ratio, or forgetting to use the …