Source: High school physics (Chinese)
Problem Sets:
Problem
A grenade moving horizontally at 10 m/s explodes into two pieces with a mass ratio of 3:2. The smaller piece flies in the opposite direction at 80 m/s.
- Find the velocity of the larger piece.
- How does the system's mechanical energy change (increase or decrease)?
[Q1] $v_1 = 70$ m/s [Q2] The mechanical energy increases.
Let the total mass be $M_{tot} = 5m$. The larger piece has mass $m_1 = 3m$, and the smaller has $m_2 = 2m$. The initial velocity is $v_i = 10$ m/s. The final velocity of the smaller piece is $v_2 = -80$ m/s. Momentum is conserved during the explosion.
[Q1] Conservation of momentum:
$$M_{tot} v_i = m_1 v_1 + m_2 v_2$$ $$(5m)(10 \text{ m/s}) = (3m)v_1 + (2m)(-80 \text{ m/s})$$ $$50 = 3v_1 - 160 \implies 3v_1 = 210$$[Q2] Compare initial and final kinetic energy.
$$K_i = \frac{1}{2} M_{tot} v_i^2 = \frac{1}{2}(5m)(10)^2 = 250m$$ $$K_f = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2 = \frac{1}{2}(3m)(70)^2 + \frac{1}{2}(2m)(-80)^2 = m(7350 + 6400) = 13750m$$Since $K_f > K_i$, the mechanical energy increases. This is due to the chemical energy released in the explosion.