March 6, 2025 |
Section: Writings
I love this simple profound proof, and it’s worth understanding. For those unfamiliar with calculus, the notation $\dot{\bf{p}}$ or $\frac{d \bf{p}}{dt}$ denotes the rate of change of $\bf{p}$ with respect to time ($t$).
We need to show that the net force ($\mathbf{F}$) applied on an object is proportional to the rate of change of that object’s momentum with respect to time ($\dot{\mathbf{p}}$).
We know Newton’s second law: $\mathbf{F} = m\mathbf{a}$, and $\mathbf{p} = m \mathbf{v}$.
August 1, 2024 |
Section: Notes
Part I: Essentials Newton’s Laws of Motion Space and Time Newton’s three laws of motion are formulated in terms of four crucial underlying concepts: the notions of space, time, mass, and force. We begin by reviewing space and time.
Space Each point $P$ of the three-dimensional space we live in can be labeled by a position vector $\bf{r}$ which specifies the distance and direction of $P$ from a chosen origin $O$. The most natural way to identify a vector is in terms of its components, which are in the directions formed by our orthonormal basis. We can do this by introducing unit vectors for each of these coordinate axes.