# How would you calculate the force of a hammer hitting an anvil? [duplicate]

If I were to hit an anvil with a hammer (let's say a 1 kg hammer at 1 m/s) how would the force of the impact be calculated?

So, where I'm at with this is that I know the deal with f = mv, so the force would be equal to 1N/s. My problem is with the time that it takes for the energy to be transferred from the hammer to the anvil. If you could see the time it takes for the collision to happen (say 0.1s) that would be easy (10N).

The hammer with the anvil is purely just an example. I've heard people mention of using a dirac delta function, but that just feels unrealistic since it's obviously not instant. Is there an inherent limit to the amount of time that it takes for the force to be applied (say the speed of sound?)

(also like to note that I have absolutely 0 idea how to use a dirac delta function. From the looks of it it's just an integral but I do not understand it)

EDIT: Thanks for the answers. I've been googling for about a week now, watching every video I could find on impulsive force, all of them say to measure it physically. You can understand why this is a very unsatisfying answer, and none of the posts answer this particular question.

Energy is not transmitted instantly. If I were to have a stick 2 lightyears long, put a button at the end of the stick, and push the stick from earth, the button would not be pressed instantly. I was curious if there's anything that determines the rate of energy transfer.

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• Can you explain your variables? I'm not familiar with the expression $f=mv$ – Aaron Stevens Nov 5 '18 at 4:30
• An illustration of how the time of contact can be estimated is shown in this example of two billiard balls colliding. The idea behind the use of a delta function is shown here. – Farcher Nov 5 '18 at 11:47
• 1. Sorry for mixing up variables. It's 3 AM. 2. I did try googling. For one week actually. I've come across everything you've linked and none of which talk about how small the timeframe can actually go. 3. Thanks. Although it doesn't really explain how to find it outside of physical measuring. – Taiga K Nov 6 '18 at 20:50

I think you mean f=ma, rather than f = mv.

The force between hammer and anvil is, indeed, very difficult to calculate. There are two significantly different cases: 1) where the interaction is elastic (no permanent deformation of either the hammer or the anvil; and 2) where the interaction is inelastic (e.g., the hammer leaves a dent in the anvil). In both cases, the force is related to the momentum change:

$$f = ma$$ or equivalently $$f = m (velocity change)/time$$

Unfortunately there is no easy way to know how long it takes for the hammer to reverse direction when it strikes the anvil. It is not quite instantaneous, because both the hammer and the anvil deform elastically when the hammer strikes.

When there is a dent, we can estimate the time by dividing the depth of the dent by the initial speed of the hammer. A typical dent is on the order of a millimeter, and a typical hammer speed at impact is probably ~ 10 meters/sec. The duration of the impact would be approximately equal to 1/10,000 sec. The force would be on the order of:

$$f = 2 kg * 20 m/sec / ((1 mm)/(10 m/sec))$$ = 400,000 Newtons.

In the first case, I think I would try to measure the force indirectly, by measuring the acceleration of the hammer. It is possible to measure the speed of the hammer head optically at nanosecond intervals. If that measurement is done continuously during impact, acceleration at each moment is approximately equal to the speed of the hammer head at the moment, minus the speed in the previous moment, divided by the time between the moments.

I don't know what the calculated force would turn out to be in the elastic case, but we know it would be greater than the force during an inelastic collision; probably a million Newtons or more.

A Dirac delta function in this context would simply mean that the speed change is instantaneous, which would mean that the force is infinite. Yes it's large, but it's not infinite!

• Yep, and this is moreover why that a hammer can rest on your arm with no problem, but if swung at you, will easily break bone. It is also the concept that some "fringe" arguers who like to make claims about how that a building cannot undergo progressive collapse because the structure below the collapsing part should be able to "hold that weight", do not get. – The_Sympathizer Nov 5 '18 at 10:25
• Thanks for this. I was wondering if it would be possible to do it outside of physical measurements, but I guess not :) – Taiga K Nov 6 '18 at 20:58