Define variable system; exploding projectile

Variable system of particles

A variable system of particles is one whose total mass changes with time because particles continuously join it or leave it, such as a rocket ejecting gas, a conveyor belt loaded with sand, or a chain being lifted from a heap. Its equation of motion is $\sum\vec F=m\vec a-\dfrac{dm}{dt}\vec u_{rel}$.

Exploding projectile

Internal forces of the explosion do not change the total momentum. Masses: $m=20$ kg, $m_A=5$ kg, $m_B=15$ kg, $v_0=30$ m/s (the numbers given in the question text; the figure shows the same arrangement in lb and ft/s).

              A (5 kg)
               /  45 deg
 v0 = 30 --->  *----------> x
               \  30 deg
              B (15 kg)

x-direction:

$$20(30)=5v_A\cos45^\circ+15v_B\cos30^\circ$$ $$600=3.5355v_A+12.990v_B\qquad(1)$$

y-direction:

$$0=5v_A\sin45^\circ-15v_B\sin30^\circ\ \Rightarrow\ v_A=2.1213v_B\qquad(2)$$

Substituting (2) in (1):

$$600=7.5v_B+12.990v_B=20.490v_B\ \Rightarrow\ v_B=29.28\ \text{m/s}$$ $$v_A=2.1213(29.28)=62.12\ \text{m/s}$$

Answer: $v_A=62.12$ m/s at $45^\circ$ above the horizontal; $v_B=29.28$ m/s at $30^\circ$ below the horizontal.