Question

A car of mass$\mathbf{2045}\mathbf{\hspace{0.33em}}\mathbf{kg}$moving in the x direction at a speed of$\mathbf{29}\mathbf{\hspace{0.33em}}\mathbf{m}\mathbf{/}\mathbf{s}$strikes a hovering mosquito of mass$\mathbf{2}\mathbf{.}\mathbf{5}\mathbf{\hspace{0.33em}}\mathbf{mg}$, and the mosquito is smashed against the windshield. The interaction between the mosquito and the windshield is an electric interaction between the electrons and protons in the mosquito and those in the windshield. (a) What is the approximate momentum change of the mosquito? Give magnitude and direction. Explain any approximations you make. (b) At a particular instant during the impact, when the force exerted on the mosquito by the car is F, what is the magnitude of the force exerted on the car by the mosquito? (c) What is the approximate momentum change of the car? Give magnitude and direction. Explain any approximations you make. (d) Qualitatively, why is the collision so much more damaging to the mosquito than to the car?

Short answer

a) The approximate momentum change of the mosquito is$0.0725\hspace{0.33em}\mathrm{kg}.\mathrm{m}/\mathrm{s}$ and it is in the positive direction, b) the magnitude of the force exerted on the car by the mosquito is also F, c) the approximate momentum change of the car is$0.0725\hspace{0.33em}kg.m/s$ and it is in the positive direction and d) the collision is much more damaging to the mosquito than to the car as the change in the momentum of the mosquito is big.

Step-by-step solution

Identification of the given data

The given data can be listed below as-

• The mass of the car is $2045\hspace{0.33em}kg.$
• The car is moving at a speed of $29\hspace{0.33em}m/s.$
• The mass of the mosquito is$0.0025\hspace{0.33em}kg.$

Significance of the law of conservation of momentum and Newton’s gravitational law on finding the momentum and collision

This law illustrates that the total momentum of a body before and after the collision becomes equal if no external forces get involved.

The gravitational law states that the force exerted on a body is directly proportional to the product of the masses and inversely proportional to the square of the distances amongst them.

The equation of the momentum of a body gives the momentum change of the car and truck and the reason for which the collision is much more damaging to the mosquito.

The equation of the force exerted on a body gives the magnitude of the force exerted on the car by the mosquito.

Determination of the change in momentum, force exerted and the collision

a) From the law of conservation of momentum, the equation of the momentum change of the mosquito can be expressed as-

$p_1=m\Delta v$

Here, $p_1$is the momentum change of the mosquito, m is the mass of the mosquito and $\Delta v$is the change of the velocity of the mosquito that is $29\hspace{0.33em}\mathrm{m}/\mathrm{s}-0\hspace{0.33em}\mathrm{m}/\mathrm{s}=29\hspace{0.33em}\mathrm{m}/\mathrm{s}$.

Substituting the values in the above equation, we get-

role="math" localid="1658207903297" $$\begin{gathered} p=(0.0025\hspace{0.33em}kg)\times (29\hspace{0.33em}m/s-0\hspace{0.33em}m/s=29m/s) \\ p=0.0725\hspace{0.33em}kg.m/s \end{gathered}$$

Thus, the approximate momentum change of the mosquito is$0.0725\hspace{0.33em}kg.m/s$and it is in the positive direction.

b) As the force exerted by the car is F that can be expressed as-

$F=G\frac{m_{\text{car}}\text{?}{m}_{\text{mosquito}}}{r^2}$

Here, r is the distance between the car and the mosquito$m_{\text{car}}and{m}_{\text{mosquito}}$

masses of the car and the mosquito respectively.

the magnitude of the force exerted on the car by the mosquito is expressed as:

$F=G\frac{m_{\text{mosquito}}\left[m_{car}\right]}{r^2}$

Thus, the magnitude of the force exerted on the car by the mosquito is also F.

c) According to the law of conservation of momentum, the momentum change of the car is the same as the momentum change of the mosquito.

Thus, the approximate momentum change of the car is$0.0725\hspace{0.33em}kg.m/s$ and it is in the positive direction.

d) Since time duration of the collision is negligible but the change in the momentum of the mosquito is big, that is the main reason the collision is much more damaging to the mosquito than to the car.

Thus, the collision is much more damaging to the mosquito than to the car as the change in the momentum of the mosquito is big.