[tex]\begin{gathered} For\text{ launcher} \\ m_L=3\operatorname{kg} \\ v_1=0\text{ m/s} \\ \text{For }projectile \\ m_P=0.08\operatorname{kg} \\ v_{P1}=\text{ 0 m/s} \\ v_{P2}=300\text{ m/s} \\ Total\text{ momentum before projectile is fired } \\ P_1=0 \\ \text{ because the system has velocity of o m/s} \\ Total\text{ momentum after projectile is fired} \\ P_2=P_1=0,\text{ beacuse of energy is constant, then} \\ \text{The momentum after projectile is fired is 0} \\ \\ For\text{ recoil velocity of the launcher }v_{L2} \\ P_2=P_1 \\ P_2=m_L\cdot v_{L2}+m_P\cdot v_{P2} \\ P_1=0 \\ m_L\cdot v_{L2}+m_P\cdot v_{P2}=0 \\ \text{Solving reco}il\text{ velocity of the launcher} \\ m_L\cdot v_{L2}=-m_P\cdot v_{P2} \\ v_{L2}=-\frac{m_P\cdot v_{P2}}{m_L} \\ v_{L2}=-\frac{(0.08kg)_{}\cdot(300\text{ m/s})_{}}{3\operatorname{kg}_{}} \\ v_{L2}=-8\text{ m/s} \\ \text{The recoil velocity of the launcher is 8m/s. The negative means the velocity is opposite of} \\ \text{the velocity of the projectile} \end{gathered}[/tex]
