There are rare chances when any insects got hit the wind shield of a car while driving in highway but it makes to wonder that how is that happen ?
Does insect suffered a greater change in momentum as compared to change in momentum of the car or the motor car was moving with larger velocity so it exerted larger force on insect or is there anything else?
Its really confusing like what's the main reason behind this ?
Best Answer
Conservation of momentum:
$$\sum p_{before}=\sum p_{after}\implies\\ p_{car,before}+p_{bug,before}=p_{car,after}+p_{bug,after}\implies\\ m_{car}v_{car,before}+m_{bug}v_{bug,before}=m_{car}v_{car,after}+m_{bug}v_{bug,after} $$
Assume the bug was still in the air before the hit, $v_{bug,before}=0$. After the hit, bug follows car $v_{car,after}=v_{bug,after}=v_{after}$.
$$...\implies\\ m_{car}v_{car,before}=(m_{car}+m_{bug})v_{after}\implies\\ \frac{m_{car}}{m_{car}+m_{bug}}v_{car,before}=v_{after} $$
All this is just showing that since the bug's mass is so much smaller than the car's $m_{car}\gg m_{bug}$, the final velocity of the car is almost not changed at all, $v_{car,before} \approx v_{after}$.
Momentum change:
The bug accelerates to the car's speed in a very short time interval. This is an enormous momentum change seen from the bug,
$$\Delta p_{bug}=p_{bug,after}-p_{bug,before}=p_{bug,after}=m_{bug}v_{after}$$
since $v_{bug,before}=0\ll v_{bug,after}$.
The momentum change of the car is very small and almost none seen from the car's perspective,
$$\Delta p_{car}=p_{car,after}-p_{car,before}=m_{car}v_{after}-m_{car}v_{before}\approx 0$$
since $v_{car,before} \approx v_{after}$. But this is only an approximation since the difference is negligible at this size-scale. In fact the momentum changes of the bug and car are exactly equal (but opposite):
$$\sum p_{before}=\sum p_{after}\implies\\ p_{car,before}+p_{bug,before}=p_{car,after}+p_{bug,after}\implies\\ p_{car,after}-p_{car,before}=p_{bug,before}-p_{bug,after}\implies\\ \Delta p_{car}=-\Delta p_{bug} $$
So, yes, they do both experience the same momentum change!
Newton's 2nd law of motion:
$$\sum F=\frac{dp_{bug}}{dt}=\frac{\Delta p_{bug}}{\Delta t}$$
Since the collision time duration is very small while the momentum change of the bug is very big, the force is very very large on the bug. The poor bug's body simply cannot withstand that force and splats.
Of course, the exact same force is exerted on the car from Newton's 3rd law. But as the car is much stronger and more massive this has extremely little effect. In the end it all comes down to the fragility and small mass of the bug - if it was a steel ball or a stone of the same size but much higher mass, the windshield migth be shattered.
In much simpler and more intuitive words think of this:
The bug is accelerated to the speed of the car; acceleration requires force, and so a large force seen from the bug's perspective is applied on the bug to cause this acceleration.
That same force applied on a car of much, much larger mass gives almost no deceleration $F=ma$.