The final velocity after the collision is 8.2 m/s
Explanation:
We can solve this problem by using the law of conservation of momentum: in fact, if we consider the system to be isolated (=no external unbalanced forces), the total momentum of the raindrop+mosquito must be conserved before and after the collision.
If the collision is perfectly inelastic, moreover, the raindrop and the mosquito stick together and travel at the same velocity v after the collision.
Mathematically:
[tex]p_i = p_f\\m_1 u_1 + m_2 u_2 = (m_1+m_2)v[/tex]
where:
[tex]m_1[/tex] is the mass of the first mosquito
[tex]u_1 = 0[/tex] is the initial velocity of the mosquito
[tex]m_2 = 50 m_1[/tex] is the mass of the raindrop
[tex]u_2 = 8.4 m/s[/tex] is the initial velocity of the raindrop
[tex]v[/tex] is the final combined velocity of the raindrop+mosquito
Re-arranging the equation and substituting, we find:
[tex]m_1 u_1 + 50 m_1 u_2 = (m_1 + 50 m_1) v\\50 m_1 u_2 = 51 m_1 v\\50 u_2 = 51 v\\v=\frac{50}{51}u_2 = \frac{50}{51}(8.4)=8.2 m/s[/tex]
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