Question 14.SP.2: A 20-lb projectile is moving with a velocity of 100 ft/s whe......
A 20-lb projectile is moving with a velocity of 100 ft/s when it explodes into two fragments A and B, weighing 5 lb and 15 lb, respectively. Knowing that immediately after the explosion, fragments A and B travel in directions defined respectively by θ_{A} = 45°~and~θ_{B} = 30°, determine the velocity of each fragment.
STRATEGY: There are no external forces, so apply the conservation of linear momentum to the system.
MODELING and ANALYSIS: The system is the projectile. After the explosion, the system is composed of the two fragments. The impulse– momentum diagram for this system is shown in Fig. 1. There are no external impulses acting on this system, so linear momentum is conserved and
m_{A}\mathbf{v}_{A}+m_{B}\mathbf{v}_{B}=m\mathbf{v}_{0}
(5/\mathrm{g})\mathbf{v}_{A}+(15/\mathrm{g})\mathbf{v}_{B}=(20/\mathrm{g})\mathbf{v}_{0}
Applying this equation in the x and y directions gives you two scalar equations. Thus,
\underrightarrow{+}x~\mathrm{components}:\qquad5v_{A}\cos45^{\circ}+15v_{B}~\cos30^{\circ}=20(100)+ ↑ y components : 5v_{A}\sin45^{\circ}-15v_{B}\ \sin30^{\circ}=0
Solving the two equations for v_{A}~and~v_{B} simultaneously gives
v_{A}=207{\mathrm{~ft}}/{\mathrm{s}}\qquad v_{B}=97.6{\mathrm{~ft}}/{\mathrm{s}}
v_{A} = 207 ft/s ⦨ 45° v_{B} = 97.6 ft/s ⦪ 30° ◂
REFLECT and THINK: As you might have predicted, the less massive fragment winds up with a larger magnitude of velocity and departs the original trajectory at a larger angle.
