# KE vs momentum

why are the equations for Kinetic energy (1/2mv^2) and momentum (mv) different? it seems to me that these are both describing the same force, if somebody could explain how they are different I'd appreciate it. Thanks.

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Neither are describing a force, what do you mean? They are more fundamental than forces. A force can be defined as the derivate of momentum.

Momentum and energy are both very important quantities, because they each have a law of conservation. You know that when you see particle change its momentum, another particle changed it also and that when you algebraically sum up the changes, you get zero. (Note that is only true in inertial frames of reference) Same goes for energy (although there is not a law of conservation of kinetic energy, unless there simply are no other forms of energy present in a specific case).

Force is a whole other thing because it factors in time in the form of a derivative and leads one to the realm of calculus.

Neither are describing a force, what do you mean? They are more fundamental than forces. A force can be defined as the derivate of momentum.
sorry, perhaps force wasn't the best term. what i mean is how can an object with a particular velocity have different measurements of kinetic energy and momentum. the most relevant definition i could find of momentum is the amount of energy it takes to stop a moving object, and kinetic energy is the energy an object has due to its motion.
won't the amount of energy an object has because it is moving and the amount of energy it takes to stop that motion be the same?
-if these are not correct definitions of KE and momentum please post better ones

This probably won't help but it got me thinking and I want to write my thoughts. Energy and momentum are the time and space components of the momentum 4-vector. Every object travels through spacetime with a velocity whose magnitude is the speed of light. In everyday situations, most objects are moving almost entirely in the time direction. Thus, changes in the direction of the 4-velocity about this extremal position results in a first order change in its space component but a second order change in its time component. That's why regular spatial momentum is proportional to v while kinetic energy is proportional to v^2. This is analagous to how a pendulum moves mainly left and right (first order) and only a little bit up and down (second order).

the most relevant definition i could find of momentum is the amount of energy it takes to stop a moving object, and kinetic energy is the energy an object has due to its motion.
That is an incorrect definition of momentum but a correct one for kinetic energy.

Momentum is the same as inertia, the tendency to stay in motion once set in motion.

Momentum is related to energy, but to know energy you must know momentum AND mass. Two objects can have the same momentum, but different kinetic energy! (p^2/2m)

There are a lot of ways to understand these basic concepts, though for most the rotation of a four-vector is probably not the easiest, hehe.

Char. Limit
Gold Member
Actually, momentum and kinetic energy are related though...

Momentum is just the derivative of kinetic energy with respect to velocity. So you could say that momentum measures the change of kinetic energy for a given velocity...

Actually, momentum and kinetic energy are related though...

Momentum is just the derivative of kinetic energy with respect to velocity. So you could say that momentum measures the change of kinetic energy for a given velocity...
At a given velocity both KE and momentum are constant and that derivative means absolutely nothing.Also mathematically you are dealing with a vector and a scalar and again it means nothing.Differentiating a scalar does not give a vector.

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Actually, momentum and kinetic energy are related though...

Momentum is just the derivative of kinetic energy with respect to velocity. So you could say that momentum measures the change of kinetic energy for a given velocity...
You mean the change in kinetic energy for an infinitesimal change in velocity.

Momentum is the same as inertia, the tendency to stay in motion once set in motion.
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Momentum is not the same as inertia. An object with velocity V=o has the same inertia as an object that is moving in a given inertial frame.Inertia is given only by mass.

Momentum is not the same as inertia. An object with velocity V=o has the same inertia as an object that is moving in a given inertial frame.Inertia is given only by mass.
Yep your right about that! Instead I will say momentum is like impulse, hehe. Just another product.

Char. Limit
Gold Member
At a given velocity both KE and momentum are constant and that derivative means absolutely nothing.Also mathematically you are dealing with a vector and a scalar and again it means nothing.Differentiating a scalar does not give a vector.
Why then, is the derivative of a scalar field, which is, after all, but a collection of scalars, give a vector?

So clearly the differential (derivative) of a scalar field must itself be a vector (it has magnitude and direction).
Right from the third paragraph.

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Yep your right about that! Instead I will say momentum is like impulse, hehe. Just another product.
This is again incorrect Impulse is the integral of force aver a time period.A change in momentum

Why then, is the derivative of a scalar field, which is, after all, but a collection of scalars, give a vector?.
A scalar field is not a scalar.A scalar field assigns a scalar to every point in space so the derivative of a scalar field being a vector field makes sense.

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I dont think I am going to keep replying to you bp. I appreciate that you want to be right, and you are, but I dont know that it is really a useful thing to say that a change in momentum is different than a momentum. Everything we deal with involves at least a reference point, e.g. in the spirit of special relativity, and in this case it would be zero momentum, or a stationary object. Then we can simply think of the momentum of a nonstationary object, i.e. momentum in general, as being imparted due to an impulse to a stationary object. In this way we can try to understand what it is to have momentum, and where it came from. I hope you can see that this type of thinking is much more useful than such a short statement that no you are wrong and my specific but very limited statement is right.

why are the equations for Kinetic energy (1/2mv^2) and momentum (mv) different? it seems to me that these are both describing the same force, if somebody could explain how they are different I'd appreciate it. Thanks.
If you wonder why there are two quantities to describe motion, you could see it this way:

Momentum has direction and magnitude and is always conserved for some reason.

Now for the special case that all of our forces are inverse square laws, you can derive that a single number mv^2/2, that you can call energy, will be conserved.

Char. Limit said:
"Momentum is just the derivative of kinetic energy with respect to velocity. So you could say that momentum measures the change of kinetic energy for a given velocity..."
bp_psy said:
At a given velocity both KE and momentum are constant and that derivative means absolutely nothing.Also mathematically you are dealing with a vector and a scalar and again it means nothing.Differentiating a scalar does not give a vector.
Constant with respect to time. But the magnitude of kinetic energy is not constant with respect to speed.

$$\frac{\mathrm{d}}{\mathrm{d}v} \left ( \frac{mv^2}{2} \right )=\frac{1}{2}\left [\frac{\mathrm{d} m}{\mathrm{d} v}v^2 + 2mv \right ]$$

which reduces to $mv$ if the mass is constant. So we could modify the statement to say that, for an object with constant mass, the magnitude of momentum is the derivative of kinetic energy with respect to speed.

I dont think I am going to keep replying to you bp. I appreciate that you want to be right, and you are, but I dont know that it is really a useful thing to say that a change in momentum is different than a momentum.
I am sorry. I did not intend to offend you with my previous post.Why I insisted on more precise definitions is because the OP is just learning about the concept and by might get confused.
Everything we deal with involves at least a reference point, e.g. in the spirit of special relativity, and in this case it would be zero momentum, or a stationary object. Then we can simply think of the momentum of a nonstationary object, i.e. momentum in general, as being imparted due to an impulse to a stationary object. In this way we can try to understand what it is to have momentum, and where it came from. I hope you can see that this type of thinking is much more useful than such a short statement that no you are wrong and my specific but very limited statement is right.
Yes. We could say that the total momentum of a particle is the maximum impulse that particle could give another particle. One issue that arises when considering different inertial frames is that momentum and KE is different in each one.This is why I like to think of momentum and KE as arising from the relation between the frames and the particles.