Why is Kinetic Energy not = mv?

[fluoronator]

I guess I'm getting old and dumb but I'm having trouble with a basic physics concept, I'm sure you can explain it...
Why is kinetic energy 1/2mV^2 and not mV?


***** Example of my confusion no1.

Suppose person A has a mass of 1 and a velocity of 10 (on ice skates).

Momentum = mV = 10.
Kinetic Energy=1/2mV^2 = 50.

Person A glides up to person B who also has a mass of 1. Person A hugs person B and they both begin to move as a unit with a total mass of 2 and speed of 5.
Momentum is conserved...
Momentum = mV = 10.

Energy is lost...
Kinetic Energy = 1/2mV^2 = 25.

Why is energy not conserved?

***** Example of my confusion no2 using the name values...

There is a body with mass of 1 floating in space (the body has a rocket engine attached). You are in spaceship A at rest relative to the body in question. I am in spaceship B with a velocity of 10 relative to you and the body. You calculate it's Ek to be 0, I calculate it to be -50.
We both watch the object accelerate so that it matches my speed, this required 1lb of fuel. Now you see me and the body traveling at a speed of 10, the body appears at rest to me. I calculate it's Ek to be 0, you calculate it to be 50.
Now it accelerates again by the same amount so that it's speed is 20 relative to you, 10 relative to me. From my perspective this again should have taken 1lb of fuel since we changed it's energy by the same amount (-50 to 0 and then 0 to 50)... but to you it took much much more fuel because you saw it's energy increase 4 times (0 to 50 and then 50 to 200).

Where is my logic messed up?

 

[jedishrfu]

With respect to 1)

There is some energy in the bonding of the pair. Normally with billiard balls ball A hits ball B, ball A stops and momentum is transferred to ball B. In your case person B hugs person A and that bond stores the missing energy.

 

[marcusl]

I guess I'm getting old and dumb but I'm having trouble with a basic physics concept, I'm sure you can explain it...
Why is kinetic energy 1/2mV^2 and not mV?


***** Example of my confusion no1.

Suppose person A has a mass of 1 and a velocity of 10 (on ice skates).

Momentum = mV = 10.
Kinetic Energy=1/2mV^2 = 50.

Person A glides up to person B who also has a mass of 1. Person A hugs person B and they both begin to move as a unit with a total mass of 2 and speed of 5.
Momentum is conserved...
Momentum = mV = 10.

Energy is lost...
Kinetic Energy = 1/2mV^2 = 25.

Why is energy not conserved?

Energy is always conserved, although it can be converted from one form to another. In this case some kinetic energy is converted to other forms of energy. The simplest example of an inelastic collision is a car crash, because you can see (and hear) where the energy goes. Energy is dissipated into heat by the collapse of energy-absorbing bumpers and the crumpling of sheet metal. Additional energy is dissipated as sound.

For your skaters, some energy is dissipated into heat as skin and tissues deform when the skaters collide, the rest is dissipated by the muscle work done when holding the skaters together.

***** Example of my confusion no2 using the name values...

There is a body with mass of 1 floating in space (the body has a rocket engine attached). You are in spaceship A at rest relative to the body in question. I am in spaceship B with a velocity of 10 relative to you and the body. You calculate it's Ek to be 0, I calculate it to be -50.
We both watch the object accelerate so that it matches my speed, this required 1lb of fuel. Now you see me and the body traveling at a speed of 10, the body appears at rest to me. I calculate it's Ek to be 0, you calculate it to be 50.
Now it accelerates again by the same amount so that it's speed is 20 relative to you, 10 relative to me. From my perspective this again should have taken 1lb of fuel since we changed it's energy by the same amount (-50 to 0 and then 0 to 50)... but to you it took much much more fuel because you saw it's energy increase 4 times (0 to 50 and then 50 to 200).

Where is my logic messed up?

I don't understand your question, but there is one obvious mistake--energy is always a positive quantity, so you can't have E_k=-50.

 

[Dale]

Why is energy not conserved?

Energy is conserved. What you should have asked is "why is kinetic energy not conserved?" The reason is because the collision is plastic which means that some of the kinetic energy is converted to other forms.

There is a body with mass of 1 floating in space (the body has a rocket engine attached). You are in spaceship A at rest relative to the body in question. I am in spaceship B with a velocity of 10 relative to you and the body. You calculate it's Ek to be 0, I calculate it to be -50.

You calculate it to be +50 J. KE can never be negative.

We both watch the object accelerate so that it matches my speed, this required 1lb of fuel. Now you see me and the body traveling at a speed of 10, the body appears at rest to me. I calculate it's Ek to be 0, you calculate it to be 50.
Now it accelerates again by the same amount so that it's speed is 20 relative to you, 10 relative to me. From my perspective this again should have taken 1lb of fuel since we changed it's energy by the same amount (-50 to 0 and then 0 to 50)... but to you it took much much more fuel because you saw it's energy increase 4 times (0 to 50 and then 50 to 200).

Where is my logic messed up?

We both agree on how much fuel it takes. You are neglecting the KE in the exhaust. For me the KE of the exhaust has less KE than for you, so the same amount of fuel was required, but the transfer of energy was more efficient than for you.

 

[Jadaav]

As the others mentioned above, energy is always conserved !

Momentum is also conserved ; Initial momentum = final momentum.

You said A glides up to B. So obviously, some of his kinetic energy will be converted to Gravitational potential energy.

When they hug each other, it's an example of collision where both body moves together with the same speed.

So when they collide, energy is converted into heat, sound, used in deformation of the bodies, and used by muscles to contract.

Therefore, there is energy lost.

Also momentum is a kind of inertia where as energy is energy.

Momentum isn't energy and so cannot be 1/2mv^2

 

[HallsofIvy]

One obvious point is that mv has units of "kg m/sec" while energy has to have units of "kg m²/t²" so kinetic energy can't be given by "mv".

 

[jedishrfu]

As a historical point, the energy vs momentum issue came up in arguments between Newton and Liebnitz:

https://eee.uci.edu/clients/bjbecker/RevoltingIdeas/leibniz.html

and I guess still resonate today.

 

[mrspeedybob]

One obvious point is that mv has units of "kg m/sec" while energy has to have units of "kg m^{2//t2" so kinetic energy can't be given by "mv".}

^{Energy has units of force x distance. If that is equivalent to what you wrote I can't see it. Could you explain it?}

 

[Bandersnatch]

Energy has units of force x distance. If that is equivalent to what you wrote I can't see it. Could you explain it?

HoI did seem to use t and sec interchangeably, but otherwise:
[N] = [kg * m/s²]
[J] = [N * m]

 

[Solkar]

Why is kinetic energy 1/2mV^2 and not mV?

p = mv
is classical momentum.

Force is momentum change
F = dp/dt,

and kinetic energy
T = mv²/2
you get by integrating force along a path.
That yields the acceleration work spent.

Pls consult Wiki or, preferably, a good introductory textbook, for more details about that.

 

[Lsos]

If you consider that Energy = Force x Distance, the result is that Energy HAS to be a square function of velocity.

For example, imagine how much braking is required to stop a car traveling at 10 m/s, and then imagine how much braking to stop the same car at 20 m/s. You apply the same brake pressure to both, and let's say the 10 m/s car takes 1 second and the 20 m/s car 2 seconds to stop. Makes sense so far. BUT, let's look at how much distance the cars cover: In the first second, the 10 m/s car covers 5 meters, and it's done. In that same time, the 20 m/s car covers 15 meters! And it STILL has another 5 meters braking left. So you see its braking distance will actually be 4x greater. 4x more braking distance is 4x more kinetic energy converted into 4x more heat.

Another way to consider it is that if you want to jump off a building to hit the ground 2x faster, you would actually have to jump from a 4x taller building. You might have 4x the distance to accelerate, but in the end gravity accelerates you only 2x longer, since the faster you go the faster you eat up the remaining distance.

 

[A.T.]

Another way to consider it is that if you want to jump off a building to hit the ground 2x faster...

In the case that the first suicide attempt failed?

 

[Lsos]

In the case that the first suicide attempt failed?

...for instance.

As for OPs confusion of "bodies floating in space", that has to do with the particularities of dealing with rockets. For one, the kinetic energy of a rocket (or anything) varies depending on what reference frame you are looking at. Since Energy = 1/2mv^2, this might give the impression that the same rocket would need to burn different amounts of fuel depending on what reference frame we look at. This is not the case, as a given fuel burn will give the rocket the same change in speed, whether it's going fast or slow. The reason for this is kind of tricky to explain, but it has to do with the fact that the rocket takes its own fuel with it, so the seemingly "free" energy it gets when its moving much faster is actually because it had to "invest" this energy into accelerating its fuel at an earlier, slower point of its journey.

Consider the point of the rocket right at liftoff, where its velocity is zero. It's still burning a ridiculous amount of fuel, but its actual energy change is also zero. Not all is lost, however, because whatever inefficiency it is suffering at the early point gets somehow "returned" at a later point, where its velocity is much greater. This is also why when doing a slingshot maneuver around a planet, its best to do it when you are closest to the planet (and traveling fastest).