# Explaining which hurts more

scenario
Put your hand with mass of $$m_0$$ on the table. Then put a brick with mass $$m_1$$ on top of your hand. Now take a hammer with mass $$m_k$$ and hit the brick at a constant speed $$v$$. now repeat the process but with a heavier brick of m2. assumes all collision is completely elastic and not heat lost or friction or any nonsense happened. Which one should hurt more. Explain.
my teachers' explain
since the force apply to the hand/brick system is the same. how much we feel only depend on the total mass of our hand/brick system. The later one(with heavier brick) is greater and thus hurt less(less acceleration).
my question

1. what exactly do we mean hurt more? more pressure? more force? more acceleration? more velocity? more energy? which one?
2. is it really justify to explain things this way? the hammer is moving at constant speed tho?
3. my explanation why heavier block hurts more is below. is this explanation better/worse.

my explanation
the initial momentum for both case is the same:

pi = v · mk (only hammer is moving)

because pi = pf && p = m·v

final speed of the whole system should be smaller than the former one(small m thus big v), and bigger for the later one(big m thus small v)

-->change in momentum in hand is greater in the former one.(both start from 0. mass didn't change. v is greater for the former one).
-->that's more energy and hurt more.
-->also, because Δ p = F · t where t is same for both scenario. the hand experience a greater force in the former example as well

feel free to analyse different scenario and possibilities. what part of the explanation is flawed?

• The term "hurts more" is purely subjective and will differ from one person to the next. In addition, this is NOT a physics term. Your teacher needs to be more specific and more precise. Commented Sep 28, 2022 at 23:27
• Both analyses are too simple for such a complex and nuanced process. I'd rather have a brick between a hammer strike and my hand, but I wouldn't want you to rest a truck filled with bricks on my hand before hitting the top with a hammer. Therefore, a heavier intermediate object isn't unequivocally good or bad in terms of reducing pain. You'd need to look at physiological nociceptor activation based on the static and dynamic loads, and that's more of a biology question. Commented Sep 29, 2022 at 0:36

This situation is a little harder to understand than we would like because it involves a collision.

Most collisions happen very quickly. Usually we don't analyze what goes on during the collision. We just look at the velocities before and after. We ask questions about conservation of momentum and energy.

But here you are asked what goes on during the collision. You put a brick next to you and hit it with a hammer. It doesn't hurt as much as if you substitute a piece of paper.

But it isn't obvious what goes on during the collision. It is over so quickly you can't see it.

So lets look at a collision. Suppose you have an electron at rest. Suppose another electron approaches it slowly enough that we can ignore magnetic forces. There is a mutual electrostatic repulsion. The force accelerates first electron so that it gains speed. The force on the second electron is equal and opposite. It loses speed. This is a slow-motion collision.

If we look at what happens during a more ordinary collision, we might see this:

This makes it clear that a collision isn't instantaneous. It is a process where two objects exert forces on each other like the two electrons.

• This collision took many microseconds. This is very quick by everyday standards.
• The golf ball goes from $$v = 0$$ to $$v =$$ fast in a very short time. $$a = \Delta v/\Delta t$$ is very large. $$F = ma$$ is very large. How much weight would you have to put on a golf ball to flatten it like that?
• You can't treat the golf ball as a rigid object that uniformly accelerates. You have to look at the accelerations of different parts of the ball.

Here is a video with more: The Moment of Impact. An Inside Look at Titleist Golf Ball R&D

We have made progress. But it still isn't obvious why a big brick protects you from these absolutely enormous forces better than a little brick. So lets look at some examples. Real Sledgehammer Smashing Bricks & Concrete in Slow Motion

Pretty much all online videos like this focus on smashing things. This isn't really what we want to see. It does drive home the point that forces are very large. But we want to see how fast the brick goes flying away. You can see that if you watch for it.

• When the hammer strikes a brick or concrete, the hammer slows down.
• Some of the energy of the hammer goes into breaking the brick. Some goes into making the fragments fly away.
• When it breaks a small brick, small pieces fly away at high speed.
• When it breaks off a big piece of concrete, the big piece moves slower.
• When the hammer pushes a big piece of concrete plus a big pile of dirt under it, the concrete plus dirt doesn't move very fast at all.

Next lets go to the classroom demo. We will use rubber bricks so the collisions are elastic. We won't hit the bricks hard enough to break them. In each case, we have two collisions:

• Hammer hits brick
• Brick goes flying and hits you

We won't focus on the deformation of the brick during the collision. The brick springs back into shape afterwards. The important things are how fast does the brick go? How much energy does it have? How much momentum?

Given the mass of the hammer and brick and such, you can work these out.

Let's take a couple examples.

• Suppose a small brick has the mass of a bullet, and winds up flying away with the speed of a bullet.
• Suppose a large brick has the mass of a truck. Suppose it winds up with the same energy, and the corresponding speed turns out to be $$0.001$$ mph.

We care a lot about the forces and deformations during the collisions when these bricks hit you. We don't have the tools to analyze this in detail, but we can get the main features.

Bullets go fast, so we expect large forces. But bullets don't slow to a stop in microseconds at the surface. The collision is more like a hammer hitting a nail. The bullet penetrates, slowing as it goes. $$\Delta v$$ goes from $$v =$$ fast to $$v = 0$$. $$\Delta t$$ is bigger than if it slowed to a stop at the surface. So forces are smaller. But they are big enough. Bullets make holes. It hurts a lot.

On the other hand if a truck hits you, the truck slows very little. You accelerate to the speed of the truck. Or even faster if you bounce off a rubber truck. This is a slow-motion collision. You go from $$v = 0$$ to $$v = 0.001$$ mph. You are squishy, so you can deform as you accelerate. $$\Delta t$$ is bigger than if you were rigid. $$a = \Delta v / \Delta t$$ is small. $$F$$ is small. It doesn't hurt.