Impulse and Change In Momentum -- Are they really different?

Question

My entire time learning physics, I have simply assumed that Impulse and Change in momentum are the same thing. It makes sense -- Force changes momentum, and impulse finds the total of force. Therefore, impulse is the total change in momentum. Why would the two ever be considered separate?

Well, apparently, my perspective is invalid! At least, according to my physics professor, and peers. As they explain it to me, impulse is "caused by force, and therefore separate to momentum. Impulse causes a change in momentum -- it isn't the change in momentum itself."

This, to me, is downright bogus. Impulse isn't force, it's caused by force. Impulse literally is the sum of the change in momentum caused by force. Impulse doesn't "cause" a change in momentum -- force causes that change! Impulse is just a fancy way of keeping track of the total amount of force applied to an object.

But here's where the ambiguity lies: "Impulse is just a fancy way of keeping track of the total amount of force applied to an object."; This can be taken to mean two things.

On one hand, it can be said to mean that impulse counts force, not change in momentum. Impulse still causes change in momentum because it is counting force, which itself causes change in momentum!

But on the other hand, it can be said to mean that impulse is the change in momentum, because force is the instantaneous change in momentum. When you sum the instantaneous changes of a value, you get an integral, and that ends up being a delta value.

Personally, I lean towards the latter of those two sides. Force really is the derivative of momentum. Force is to momentum as acceleration is to velocity, and this is painfully obvious since all one needs to do is multiply acceleration and/or velocity by mass. And so, saying that impulse is not the same as change in momentum, and that impulse is actually just a type of force is like saying that the integral of velocity over time is not the change in position, and that the integral of velocity over time is another type of velocity. It just doesn't make sense to me!

Now, often times when I make this argument to people, they immediately jump to the conclusion that I think the two are the same because they always have the same value, and because mathematically they are the same. Personally, I think that mathematically proving that two things are the same is enough to say that they are the same. The "2* pi * r" of a circle is the same as the circumference of that circle, because mathematically, they have been proven to be the same. But, quite frankly, even ignoring the fact that, mathematically, impulse and change in momentum are the same, I still think that conceptually they are the same, and I think that for the reasons given above.

So here, really, lies my question: Is there even a point to arguing about this? Is this a controversial topic in physics? Surely I am not the only one who thinks that these two concepts ought to be treated as one? So far, no argument that has been presented to me has been enough to convince me that it is practical, or even conceptually enlightening to keep them separate. If anything, I think it is beautiful that they are the same, and that considering them to be separate actually prevents a deeper conceptual understanding of the universe and mathematics as a whole. Am I crazy for thinking that?

Answer 1

So here, really, lies my question: Is there even a point to arguing about this?

Perhaps there is a point in discussing this.

In the Newtonian point of view, impulse and change of momentum are different concepts. Why?

Force $F(t)$ is a basic quantity describing instantaneous influence of one body on another, in general having a magnitude and direction, but let's have everything in the same direction here for simplicity. The formula for force has to be inferred from other laws of physics - it can be due to gravity ($mg$), spring ($-kx$), or air resistance $(-cv^2)$ or others or their combination.

With this notion of force, impulse of the force $F$ in the time interval $t_1..t_2$ is defined as $$ I = \int_{t_1}^{t_2} F(t) dt. $$

One could calculate impulse without knowing anything about momentum.

Now, based on the 2nd law for body with constant mass (not definition here)

$$ F = m\frac{dv}{dt}, $$ we can derive that $$ I = m \Delta v $$ and since $m$ is constant, also $$ I = \Delta (mv). $$ which means that impulse equals change in momentum. This is the historic and common point of view, I believe.

Alternatively, if you take the point of view where "force" is defined as $ma$, then impulse and change of momentum of the body have the same values as a consequence of definitions only. But I wouldn't say this means that impulse and change of momentum are the same concepts, because they are introduced in a different way with different name and symbol. So either way, I think it is safe to say that both are different concepts, while having the same value, either approximately (if 2nd law is taken as approximate law of physics) or exactly (if it is taken as a definition of force).

Answer 2

Force really is the derivative of momentum.

Except that it isn't. Have you taken a Statics course yet? Don't forget, in the equation

$$\vec F = \frac{d\vec p}{dt} $$

the left hand side is understood to be the (vector) sum of all forces acting on the particle, i.e., it is the net force

$$\Sigma \vec F = \frac{d\vec p}{dt}$$

Also, there are alternate formulations of mechanics, e.g., Lagrangian mechanics, in which the notion of force is absent and, rather than $F = ma$, we have the Euler-Lagrange equation

$$\frac{d}{dt} \frac{\partial L}{\partial \dot x} = \frac{\partial L}{\partial x} $$

where the Lagrangian $L$ is the difference of the particle's kinetic and potential energy.

$$L = T - V$$

For example, for a mass on a spring system, we have

$$L = \frac{1}{2}m\dot x^2 - \frac{1}{2}kx^2$$

and the Euler-Lagrange equation yields

$$m\ddot x = -kx$$

We recognize the left hand side as the time rate of change of momentum and the right hand side as the (negative of the) spatial rate of change of the potential energy.

But, on the Newtonian view, the right hand side is the force on the mass due to the spring, not the left hand side.

So, I don't think one can successfully sustain "Force really is the derivative of momentum."