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The key things to remember are:

1.) $F = ma$

2.) $a = \frac{\mathrm{d}v}{\mathrm{d}t}$

For a $100~\text{kg}$ man standing on the nail: $F = 100~\text{kg} \cdot 9.8~\frac{\text{m}}{\text{s}^{2}} = 980~\text{N}$.

For a $\frac{1}{2}~\text{kg}$ hammer head, swung at $10~\frac{\text{m}}{\text{s}}$: $F = 0.5~\text{kg} \cdot a = ?~\text{N}$.

$a$ in this last equation is the deceleration of the hammer head when it hits the nail. Let's say that the hammer drives the nail $x = 2~\text{mm} = 0.002~\text{m}$ with each stroke, and further assume that the deceleration of the hammer head is constant (makes the math easier). Then you get the quadratic:

$t^{2} - \frac{20}{a}t + \frac{4}{1000a} = 0$

Substituting $a = \frac{10~\frac{\text{m}}{\text{s}}}{t}$ into the equation $t = \sqrt{\frac{2x}{a}}$, we get $t = 0.0004~\text{s} = 0.4~\text{ms}$. If we use that $t$ in the quadratic we find that $a = 19060~\frac{\text{m}}{\text{s}^{2}}$.

So $F = 0.5~\text{kg} \cdot 19060~\frac{\text{m}}{\text{s}^{2}} = 9530~\text{N} \implies$ roughly $10$ times the force of standing on the nail.

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Of all 10 answers to this question and its duplicate, this is by far the best. – alx9r Feb 9 at 2:57


First, let me be sure to mention several quantities.

  • Impulse ($I=m v$)







  • The mass of the hammer head
  • The material stiffness of the nail ($k$)
  • The height it falls from




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To drive a nail into a piece of wood you need to overcome the force of static friction and the force required to push aside the wood (make a hole).

When an object of mass $m$ and velocity $v$ hits a nail, either the nail moves, or the object decelerates very quickly. This sudden change in momentum is what drives the nail. We know that

$$F\Delta t = m\Delta v$$

So if you want to get a greater force you can change any of these parameters:

  • increase the mass (heavier hammer)
  • move faster (hit harder)
  • shorter $\Delta t$

The latter is a function of the elasticity of the hammer and the nail: as the nail is thicker, or less is sticking out of the wood, it will be a stiffer "spring" and deform less during the impact. This means the hammer will exert a greater force. This is one reason why you can keep hammering a nail as it goes deeper into the wood: while there may be more force needed, the shorter nail provides a greater "force amplifier", in the form of shorter $\Delta t$.

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