How can I calculate the force a giant knife applies to a car if it cuts straight through?

I feel like this is a really basic question but I'm struggling to come up with a straightforward answer!

In this video:

We see a 10 foot knife dropped from 30 feet onto a car. It pierces straight through the car and only stops when it hits the ground.

I can calculate the easy stuff, such as the gravitational potential energy of the knife before being dropped, its velocity and momentum at the moment it reaches the car, impact force on the ground etc. but I need to explain how it's able to pierce so easily through the car.

I believe it's because of the pressure applied by the point of the knife, but I can't seem to calculate anything to do with force applied to car or energy transfer from the knife to the car, as it doesn't look like it slows it down at all really.

I've looked a bit at the physics of ballistics but can't find anything that applies to this situation. Can anyone help?

I'd quite like to be able to work out how many cars the knife would go through if it weren't for the ground, but I feel like I'd need quite a lot of information to calculate this, e.g. the density of the car's body, and the knife, area and thickness of the knife et cetera.

I don't know anything at all about the 'physics of cutting' so I am at a bit lost!

• I think without a lot more information on the car's material properties along with the knife's material properties, anything you determine would be highly approximate and not very useful. – JMac Aug 16 '17 at 10:38

So there are a few things you need to know. First is what the car is made out of. A quick google told me that "The typical family sedan these days is usually made from steel body panels with plastic front and rear fascias". And since it's piercing through the top we can assume that it's going through the steel. Another thing you'll need to know is the weight of the knife, and roughly what the shape is. Using the equation $$\mathrm{Pressure} = \mathrm{\frac{Force}{Area}}$$ you can work out the pressure that the knife exerts on the car. Working out the force at this point is easy, as we know the gravitational field strength of the earth is $9.81ms^{-2}$ so all you need to do is multiply that by the mass. The area will be harder if you want $100\%$ accuracy, as it changes as the knife goes into the car, however we can approximate this to a rectangle of dimensions equal to $\mathrm{width \times maximum\ length}$

From this point you'll need to know the pressure required to pierce steel, which can be easily searched on the internet.

Here comes the hard part. You will need to know the energy lost in the impact and due to friction. Energy lost from friction is equal to the work done from the friction force. For this you will need to use the equation $$F_{max} = \mu R$$ where $F_{max}$ is the maximum friction (which it will be as they are moving against each other), $\mu$ is the coefficient of friction and $R$ is the reaction force. $\mu$ should be relatively easy to find, assuming you know the material of the car and the knife. The reaction force is just equal to the force pushing down on the car. Energy does not need to be explicitly calculated but can be included by considering change in momentum.

Once you've calculated the force on the first car by the knife with $$F = \mathrm{ma}$$ then you can no longer use this. You'll have to use $$F = \frac{\mathrm{Change\ in\ momentum}}{\mathrm{time}}$$You can account for this by approximating how fast the car will move upon impact and then taking into account that momentum is conserved.

Change in momentum is calculated from $\mathrm{m}(v-u)$ and the final speed ($u$) can be calculated from the change into momentum and taking friction into account with the energy loss. Hope this helps :)

• "From this point you'll need to know the pressure required to pierce steel, which can be easily searched on the internet." You might be able to find rough approximations; but it will depend on more than just pressure. Cutting through metal with a sharp object is different than busting through it with a blunt object. Not just in terms of the force distribution; but also how the material fails may be quite different in each scenario. – JMac Aug 16 '17 at 10:30
• Amazing thank you! Everything you say I understand, for some reason I was just struggling to put it all together into something useful...this makes sense to me though. – E Manners Aug 16 '17 at 11:23
• Although as JMac says, there are a lot of variables here, and any calculations will be extremely rough... – E Manners Aug 16 '17 at 11:55