# 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?

• Did you mean "free fall without friction" in the second line? – SchrodingersCat Nov 25 '15 at 11:58
• no with friction, a falling stone with a limit speed. But it doesn't make any difference, the point is that the stone goes with a constant speed – engineer_abc Nov 25 '15 at 12:12
• let $t_2$ - $t_1$ = $dt$, So from $t_1$ to $t_2$ there is an average acceleration (or retardation) 'a' which changes the velocity of the ball. The acceleration is not infinite. – manshu Nov 25 '15 at 12:37
• YOu've just discovered Green's Function, more or less. Take the limit of decreasing time and increasing delta velocity, and you'll discover the $\delta$ function. – Carl Witthoft Nov 25 '15 at 12:43

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.
I am only going to leave a brief answer, seeing that the comments are very accurate. The paradox can simply be resolved by considering the elastic nature of all the objects. How so ever instantaneous might the $dt$ or the time of collision seem to the human eye, actually it occurs over a small duration, based on the elasticity of both the objects involved in the collision. Thus the $dt$ never actually reaches zero and the acceleration never actually reaches infinity, where in principal it is free to do so.