Sunday, April 10, 2016

newtonian mechanics - Is the reaction force for a stone hitting a wall infinite?


Let us assume a rigid stone which moves in empty space with a constant speed of $v$. (Or in the air with no friction and drag or you can imagine a free fall with friction).


This stone hits a rigid wall and stops or goes back with a constant speed. If we analyze the very moment that stone hits the wall, the acceleration of the stone decreases tremendously in a very small amount of time since the speed is decreasing instantaneously. We can consider the amount of time as "$\mathrm{d}t$" since the time is infinitesimally small. If the crash happens in a very small amount of time $\mathrm{d}t$, then the speed will decrease in a very small amount of time which means a huge negative value of acceleration (or deceleration). By Newton's second law $F=ma$, the force is always equal to acceleration times mass. If we consider the time interval as infinitesimally small, i.e. $t\to 0$, then the acceleration will be infinitely high and negative. And this infinite value makes the force infinitely great, so that the very moment which the stone hits the wall, the exerted force will be infinite.


Even though this incident seems to happen in an instant moment, does that make sense? An infinite force has to create a massive energy. But the reality isn't so. Then how come we explain the incident with the Newton's second law?



Answer



It is a hypothetical condition as inertial will never let this condition happen. For the sake of argument I am using impulse. faster you stop an object more will be the force. Example using gloves to stop a fast ball in sports. $$F_{impact}*t=mv-mu$$ $$F_{impact}=\frac{mv-mu}{t}$$ $$F_{impact}=\lim_{t \to 0}\frac{mv-mu}{t}$$ According the equation the force will be infinite which important to remember this is hypothetical.


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