Thread: Velocity and kinetic energy

Can anyone explain the relationship between velocity and kinetic energy?

From what I understand, when you double the velocity of an object the kinetic energy is increased by a factor of four. This seems counter-intuitive to me.

Let v be the velocity of a particle measured in some inertial reference frame and for ease of notation lets suppose that one of the coordinate axis are parallel with the velocity vector (i.e. so v = vi for example) . Then the kinetic energy of the particle is KE = mc² - m₀c² which for small velocities gives the equation KE ~ 1/2 mv². So if you double the velocity of the particle then KE ~ 1/2 m(2v)² = 4*(1/2 mv²)

In other words doubling the velocity increases the kinetic energy by a factor of 4 if the velocities are sufficiently small.

I wish my math skills were better.

Could you please expalin what the m0c^2 represents in the equation KE = mc^2 - m0c^2.

einsteins famous equation wich everybody know E=M₀C²
C is the velocity of light, about 3*10⁸
and M₀ is the mass of the object at 0 velocity to a referensframe. and M is the mass of the object when its moving at velocioty v
M=M₀/sqr(1-v²/c²)

Got it, thanks Zelos.

no problem
You can rewrite the equation like
Ek=M₀C²(1/sqr(1-v²/c²)-1)

Originally Posted by DarcgreY

Can anyone explain the relationship between velocity and kinetic energy?

From what I understand, when you double the velocity of an object the kinetic energy is increased by a factor of four. This seems counter-intuitive to me.

Why does it seem counter intuitive?

I personally used to think that too but have come to accept it. Yet it has never seemed intuitive .... then again if it was linear it wouldn't be intuitive either. Hmmmmm ... more questions than answers for me here.

Pete

Common sence isn't going to tell you that if you double the velocity of an object the kinetic energy is going to be four times. Early scientists like Galileo were suprised when they discovered this relationship through experiments.

Originally Posted by DarcgreY

Common sence isn't going to tell you that if you double the velocity of an object the kinetic energy is going to be four times. Early scientists like Galileo were suprised when they discovered this relationship through experiments.

yes but later discoveries were even more supricing that it really isnt a double of speed equals 4x energy but that it increases even faster wich is more noticble at NCV

It is common sense, Darcgrey, you just have to see it right. If you whack a baseball and get it going at 20m/s, you've got to run after it and hit it again to get it going at 40m/s. That running after it is a whole pile of extra work. Here's an rehash from ENERGY EXPLAINED that tells you more:

Consider a 10 kilogram cannonball, in space, travelling at 1000 metres per second. We talk about how much kinetic energy this cannonball has. We talk about KE = ½ MV2 and we do the maths and get five million Joules. But what has the cannonball really got? Its mass seems real enough, and its motion seems real enough too.

To find out more I take a spacewalk to place a thousand sheets of cardboard in the path of my cannonball. Each sheet of cardboard exerts a small braking force, slowing the cannonball to a halt in two seconds. We know that the cannonball will punch through more cardboard in the first second than in the second second, because it’s slowing down. So we deduce that a cannonball travelling at 1000m/s has more than twice the kinetic energy of one travelling at 500m/s. We can do the arithmetic for each second, then slice the seconds up finer and finer, and we end up realising that the ½V2 is the integral of all the velocities between V and 0.

The kinetic energy is a way of describing the stopping distance for a given force applied to a given mass moving at a given velocity. You can flip it around to think about force times distance to get something moving. Or you can think in terms of damage. But basically that cannonball has “got” kinetic energy like it has “got” stopping distance.

It’s similar with momentum. That’s a different way of looking at the mass and the motion, based on force and time instead of force and distance. We look back to our cannonball and cardboard, and we know by definition that in the first second the same amount of time passed as in the second second. So we realise that a cannonball travelling at 1000m/s has twice the momentum of one travelling at 500m/s.

The momentum is a way of describing the stopping time for a given force applied to a given mass moving at a given velocity. Again you can flip it around, but basically that cannonball has “got” momentum like it has “got” stopping time. Note that momentum is always conserved in a collision because two objects are in contact for the same length of time.

Originally Posted by Farsight

So we realise that a cannonball travelling at 1000m/s has twice the momentum of one travelling at 500m/s.

Isn't the relationship exponential not linear. That's what I was talking about when I said I thought it was counter-intuitive. A cannonball traveling at 1000m/s has four times the momentum of one traveling at 500m/s doesn't it?

It's twice the momentum, but four times the kinetic energy.

Momentum is force times time, so if you stop the 1000m/s cannonball in two seconds with a constant force, in the first second it loses half of its momentum, and in the second second it loses the other half.

Kinetic energy is force times distance. In the first second its average velocity was 750m/s, while in the second second its average velocity was 250m/s, so it covered more distance in the first second than in the second second. So it lost more Kinetic Energy in the first second than in the second second.

Momentum is force times time,

yes, but the defintion is momentum is mass times velocity
force is defined as delta momentum divided with delta time

The definitions are a tad circular, Zelos. And a photon's got momentum but no mass. Well, that's what they say, I say it's got mass, only I'll get into trouble with the definitions there too. The thing is that inertia is like momentum pinned down into one place. See MASS EXPLAINED for more.

what it is and what defined it is 2 different things

This is what I was on about. It's all good stuff:

http://www.thescienceforum.com/MASS-EXPLAINED-5069t.php

When we turn our attention from a cannonball to a photon, we have to express the energy and the momentum in a different way. There is no “mass”, so the energy is hf, and the momentum is hf/c. The h here is Planck’s constant of 6.63 x 10-34 Joule-seconds, and is an “action” which is a momentum multiplied by a distance. The f is the frequency per second, and our old friend c is distance over time, which converts a stopping-distance measure into a stopping-time measure. It’s just λ/c or wavelength over frequency, so you can also express the momentum as h/λ. And you can see how that momentum affects a mass via Compton scattering...

what about it?
Why photon can have momentum is cause energy and mass is the same so if something have energy it has momentum aswell