# Conservation of Mechanical Energy Before and After Impact of a Hammer

If a hammer does work by driving a nail into a wooden board, how does the mechanical energy from right before the hammer hits the nail compare to the mechanical energy after the nail has been driven into the board?

Since mechanical energy is conserved, shouldn't it be the same before and after impact? I'm confused because if the hammer does positive work, it has lost kinetic energy. Before impact, it has some non-zero velocity $v_i.$ After impact, $v_f = 0,$ so the final kinetic energy is less than the initial kinetic energy, which indicates a decrease in mechanical energy. Doesn't this conclusion contradict the conservation of mechanical energy?

• I have never heard of the conservation of mechanical energy as a law of nature
– user83548
Commented Nov 13, 2015 at 5:01

There are different forms of energy. Energy can be converted from one form to another but cannot be destroyed. In this case the kinetic energy of the hammer is driving the nail into the wood which is breaking the molecular bonds in the wood fiber. The energy is converted to heat energy as a result of the breaking of the bonds and the friction of the nail in the wood.

• That's what I suspected as well. There must be a change in some other form(s) of energy that counteracts the decrease in kinetic energy. But ultimately, mechanical energy is conserved, so its initial state is equal to its final state, right? Commented Nov 13, 2015 at 5:09
• conservation of mechanical energy is not a law of nature, but conservation of energy is. Some of the mechanical energy is converted into heat, which rises temperature, which is a rise in internal energy at the expense of mechanical energy.
– user83548
Commented Nov 13, 2015 at 5:13
• Okay, so then mechanical energy decreases. But then when is mechanical energy conserved? Commented Nov 13, 2015 at 5:19
• when the forces involved are conservative, Read: en.wikipedia.org/wiki/Conservative_force
– user83548
Commented Nov 13, 2015 at 5:27

Conservation of Mechanical Energy Before and After Impact of a Hammer

The kinetic energy of a hammer of mass $m$ ans speed $v$ is equal to the work done by the resistance force , $F_r$ the nail faces while it travels the distance $d$ in the wood.

$mv^2 /2 = F_r * d$

$F_r * d$, like any work done by a friction force, converts mainly into heat.

Conservation of energy law applies to a system, not to one component of it. So in this case it applies to hammer+nail system, which can be expressed in terms of momentum too : $$p_{h0} + p_{n0} = p_{h1} + p_{n1}$$

where subscript $$0,1$$ indicates body momentum before and after an event has occurred (hammer impact this time)

Or to be expressed in general, system's total momentum stays the same :

$$\sum_i{p_i} = \textrm{const}$$

So hammer has transferred it's momentum to nail and nail has penetrated board until friction forces has stopped it.

EDIT:
Some kinetic energy may be dissipated into heat, also it depends on collision type - elastic / inelastic.
At other times kinetic energy may be converted to potential energy - and that's a different conservation law :

$$E_k + U = \textrm{const}$$

Now this law may be applied to one body - consider a ball which you throw straight vertically into the air. Then ball velocity decreases, until it stops at some altitude, but at that point it will have maximum potential energy $$U_{max}$$. Then it will start to go back to ground, then again converting it's accumulated potential energy back into kinetic. And so on, until ball's kinetic energy will be dissipated upon impact with ground.

Hope that helps.