Relation between Impulse and Conservation of Momentum

[jstad]

First post here. This question has two parts. (1) Connecting the dots between the Impulse-Momentum Theorem and the Law of Conservation of Momentum and (2) Book recommendations for a more theoretical treatment of classical mechanics?

(1) Difficulty reconciling the Impulse-Momentum Theorem with the Law of Conservation of Momentum.

Am I over-thinking this concept? Is it really just that the Impulse-Momentum Theorem shows that Newton's 3rd law holds in a closed system and that when objects interact (e.g., collide), the change in their momentum is F_net = 0?

I suppose my difficulty is that instantaneous change in momentum, dP/dt ≠ ∫F(t)dt. Instead, ∫F(t) = ΔP. ∫F(t)dt produces kg m/s while dP/dt gives us kg m/s².

Therefore, the area underneath the Force/Time curve simply gives us the change in momentum, while the Law of Conservation of Momentum uses the derivative (instantaneous change in momentum) to show that internal forces cancel out in a closed system and thus momentum is conserved.

I am confused on how one makes the leap from the Impulse-Momentum Theorem to the Law of Conservation on Momentum. It seems to me that the Law Of Conservation of Momentum should be based on the idea of Impulse, yet the units don't add up (Impulse = N s; dP/dt = N). It seems that implicit in the Impulse is the idea that momentum is conserved, i.e., P_i = P_f. But again, the units don't add up between the two concepts. How do these two ideas relate/reconcile with one another?

(2) I am working out of Knight's "Physics: For Scientists and Engineers, 3rd Edition," pp. 221 - 229. I'm pretty sure the theoretical side of the calculus is my problem. Yet this book does nothing to connect the dots. I think it's more designed for the plug and chug world of engineering. With that said, are there any books that provide a more theoretical, yet accessible side to classical mechanics? Has anyone worked through Spivak's "Physics for Mathematicians"? That title seemed appealing to me.

Thank you in advance!

 

[jtbell]

I think of the impulse-momentum theorem as being basically Newton's Second Law, expressed in integral form. Newton's Third Law is separate. You need both of them to get conservation of total momentum.

Consider two colliding objects. During the collision, object 1 exerts force $\vec F_{12}(t)$ on object 2. Object 2 exerts force $\vec F_{21}(t)$ on object 1.

According to the impulse-momentum theorem (Newton's Second Law), objects 1 and 2 change momentum by
$$\Delta \vec p_1 = \int_{t_1}^{t_2} {\vec F_{21}(t) dt}\\
\Delta \vec p_2 = \int_{t_1}^{t_2} {\vec F_{12}(t) dt}$$

According to Newton's Third Law, $\vec F_{21}(t) = - \vec F_{12}(t)$. It follows from the above that $\Delta \vec p_1 = - \Delta \vec p_2$.

Therefore the change in total momentum is $\Delta \vec p = \Delta \vec p_1 + \Delta \vec p_2 = 0$.

 

[jstad]

Thank you for your reply. This is completely intuitive and clear to me. I think my book may have introduced dP/dt so as to directly relate Conservation of Momentum to Newton's 2nd Law (the F = ma incarnation). I will post if I have any further questions but thank you again for clearing this up. The way the book presented this was very anti-intuitive to me.