An intuitive understanding of momentum and kinetic energy

[hokhani]

Suppose that we have two balls (1) and (2) with the masses m1 and m2 and velocities v1 and v2, respectively. Furthermore, suppose that their momentums and kinetic energies are not the same so that P1>P2 and K1<K2. Which ball is more dangerous in hitting a person.

 

[hutchphd]

This is a nicely phrased question.
The answer depends upon the details of the collision in ways that probably an ER physician is best equipped to answer. That being said the most obvious question is how "elastic' is the collision. If the balls bounce off elastically then all the energy is returned to the balls and presumably little irreversible damage is done. If the balls embed then the inelasticity implies permanent deformation. The demarcation between these regimes depends crucially on factors that include the physical size of the balls and the speeds involved.
So there is no simple answer.?

 

[hokhani]

This is a nicely phrased question.
So there is no simple answer.?

Thanks. Suppose that the two balls stop after a millisecond of their strikes. I would like to know which factor is determining in damage; transfer of momentum or energy. In other words, the ball which transfers larger momentum is more dangerous or the ball which transfers larger energy?

 

[jrmichler]

If you shoot your eye out with a Daisy Red Ryder BB gun (0.17" diameter steel BB weighing 5 grains at 270 ft/sec), would you really care?

You can answer your own question: Think of some real world objects. How fast is a baseball? What would happen if one hit you? Calculate the kinetic energy and momentum. How fast is somebody running? What would happen if they ran into you? Calculate the kinetic energy and momentum. Repeat for other objects until you see a pattern.

 

[hutchphd]

Thanks. Suppose that the two balls stop after a millisecond of their strikes. I would like to know which factor is determining in damage; transfer of momentum or energy. In other words, the ball which transfers larger momentum is more dangerous or the ball which transfers larger energy?

Let me assume that the maximum force delivered is the destructive factor. ...that seems reasonable.
Then if the balls stop over a fixed time the one with the largest momentum will deliver biggest force .
Notice if they stop over a fixed distance then the one with the largest energy will deliver the biggest force.
I think the second case sounds more realistic.

Other comments notwithstanding I find the question interesting.

 

[A.T.]

Which ball is more dangerous in hitting a person.

You can kill a person with minimal mechanical damage. For example, by rupturing a blood vessel at the brain through acceleration of the head. If you are interested in physics not biology, I suggest using a simpler body.

Thanks. Suppose that the two balls stop after a millisecond of their strikes.

Stop and remain at rest relative to the target like in a completely inelastic collision?

 

[hokhani]

Let's consider a spring instead of a person. When a ball hits on a spring, it compresses the spring until the spring stops instantly. At this instant of time, all the energy of the ball is transferred to the spring and also a mean force equal the momentum of the ball transfers to the spring.
Now, the situation becomes more interesting;
i) On one hand, we expect the ball (1) with greater momentum transfers greater force to the spring and so the spring becomes more compressed.
ii) On the other hand, the energy of the ball (1) is less than (2) and so we expect that the ball (1) compresses the spring less than the ball (2). It seems a discrepancy between the cases (i) and (ii)!

 

[jbriggs444]

i) On one hand, we expect the ball (1) with greater momentum transfers greater force to the spring and so the spring becomes more compressed.

"Force" is not something that is transferred. Momentum is transferred. However, you are correct that the ball with greater momentum will transfer more momentum to the spring. This makes sense since, all other things being equal, the ball with greater momentum (lower speed but higher mass) will spend more time compressing the spring. The momentum that is transferred is the integral of force over time.

The conclusion is incorrect, however. 100% of the momentum transferred to the spring is transferred on out to its anchor point. None of it is retained. It is irrelevant to the compression of the spring.

ii) On the other hand, the energy of the ball (1) is less than (2) and so we expect that the ball (1) compresses the spring less than the ball (2). It seems a discrepancy between the cases (i) and (ii)!

Yes. This one is correct. The work done by the ball on the spring relates to the distance the spring is compressed. The energy that is transferred (the work done) is given by the integral of force over distance. More energy, more work, more distance. This time no energy is transferred out the far end of the spring. The anchor point does not move. The distance moved is zero, so the work done is zero.

 

[hokhani]

"Force" is not something that is transferred. Momentum is transferred. However, you are correct that the ball with greater momentum will transfer more momentum to the spring. This makes sense since, all other things being equal, the ball with greater momentum (lower speed but higher mass) will spend more time compressing the spring. The momentum that is transferred is the integral of force over time.

The conclusion is incorrect, however. 100% of the momentum transferred to the spring is transferred on out to its anchor point. None of it is retained. It is irrelevant to the compression of the spring.

Yes. This one is correct. The work done by the ball on the spring relates to the distance the spring is compressed. The energy that is transferred (the work done) is given by the integral of force over distance. More energy, more work, more distance. This time no energy is transferred out the far end of the spring. The anchor point does not move. The distance moved is zero, so the work done is zero.

With your statements, we can infer that it is the energy (not the momentum) which determines the force exerted by the ball. So the more energy, the more is the exerted force. But I still don't have any intuitive picture of the momentum.

 

[jbriggs444]

With your statements, we can infer that it is the energy (not the momentum) which determines the force exerted by the ball. So the more energy, the more is the exerted force.

More energy, more total displacement, more peak force. One could reason that way.

But "peak force" is not the be all and end all of characterizing an impact.

 

[vco]

I like to think that energy is a "fundamental quantity" and momentum is just a handy rephrasing of Newton III.

Generally, energy does the damage. But the real world is more complicated than that.

 

[gleem]

The energy in the center of mass will be the energy available to produce any damage in the colliding bodies.