Question

A hovering mosquito is hit by a raindrop that is 50 times as massive and falling at 8.4 m/s, a typical raindrop speed. How fast is the raindrop, with the attached mosquito, falling immediately afterward if the collision is perfectly inelastic?

Answer 1

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:

$p_i = p_f\\m_1 u_1 + m_2 u_2 = (m_1+m_2)v$  

where:  

$m_1$ is the mass of the first mosquito

$u_1 = 0$ is the initial velocity of the mosquito

$m_2 = 50 m_1$ is the mass of the raindrop

$u_2 = 8.4 m/s$ is the initial velocity of the raindrop

$v$ is the final combined velocity of the raindrop+mosquito

Re-arranging the equation and substituting, we find:  

$m_1 u_1 + 50 m_1 u_2 = (m_1 + 50 m_1) v\\50 m_1 u_2 = 51 m_1 v\\50 u_2 = 51 v\\v=\frac{50}{51}u_2 = \frac{50}{51}(8.4)=8.2 m/s$

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