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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?

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  • $\begingroup$ Did you mean "free fall without friction" in the second line? $\endgroup$
    – SchrodingersCat
    Commented Nov 25, 2015 at 11:58
  • $\begingroup$ 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 $\endgroup$
    – engineer_abc
    Commented Nov 25, 2015 at 12:12
  • $\begingroup$ 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. $\endgroup$
    – manshu
    Commented Nov 25, 2015 at 12:37
  • $\begingroup$ 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. $\endgroup$
    – Carl Witthoft
    Commented Nov 25, 2015 at 12:43

2 Answers 2

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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.

This is also used in various places, for instance ball game players move their hands as they catch the ball to maximize the period of impact.

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    $\begingroup$ If you consider the comments to be accurate, I encourage you to incorporate them into your answer. After all, those comments could easily disappear someday; your answer won't (unless you delete it). $\endgroup$
    – David Z
    Commented Nov 25, 2015 at 13:09
  • $\begingroup$ So, if the stone and the wall are ideally rigid (not elastic), then is it true that the reaction force will be infinitive in an instant of time? $\endgroup$
    – engineer_abc
    Commented Nov 25, 2015 at 14:26
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    $\begingroup$ @engineer_abc - "ideally rigid" would also imply infinite speed of sound, which violates relativity (information about the collision would reach all points in the stone at the same time). If you want to leave the realm of physics, by all means do. But it might be a lonely journey. $\endgroup$
    – Floris
    Commented Nov 25, 2015 at 14:59
  • $\begingroup$ @engineer_abc yes, the force would be infinite, but they should also be infinitely strong, otherwise they would simply break. $\endgroup$
    – Hemang Rajvanshy
    Commented Nov 25, 2015 at 15:10
  • $\begingroup$ @engineer_abc you can't describe rigid body physics with force. You use the impulse of force (force integrated over time) instead. Then change in velocity = impulse of force / mass. Then decreasing the elasticity of a collision increases the force and decreases the collision time but the product is preserved. $\endgroup$
    – John Dvorak
    Commented Nov 25, 2015 at 16:18
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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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