Problem with resources, Walter Lewin's third lecture

Question

I've watched Walter's third lecture in 8.01 and I have a small problem with the last part, where he says that $$\vec r_t=x_t\cdot \hat x\ +\ y_t\cdot \hat y\ +\ z_t\cdot \hat z \\ \vec v_t=\frac{d\vec r}{dt}=\dot x \hat x\ +\dot y \hat y\ +\dot z\hat z\\ \vec a_t=\frac{d\vec v}{dt}=\ddot x \hat x\ +\ddot y \hat y\ +\ddot z\hat z$$ and I have NO problem in understanding that, at least intuitively, but where can I find a resource to read about that in detail? a book that's only prerequisite would be calculus 1, I need some exercises on that topic and I can't really find something like that in Giancoli

Answer 1

I don't think you really need a rigorous proof here. I mean the first one is the definition of position and then to get the others you're just taking the derivative and applying linearity and the fact that for cartesian coordinates the basis vectors are constant even if the object in question changes position. $$\frac{\mathrm{d}}{\mathrm{d}t}(f+g+\ldots)=\frac{\mathrm{d}f}{\mathrm{d}t}+\frac{\mathrm{d}g}{\mathrm{d}t}+\ldots \\ \frac{\mathrm{d}}{\mathrm{d}t}(x(t)\hat{x})=\frac{\mathrm{d}x}{\mathrm{d}t}\hat{x}+x\frac{\mathrm{d}\hat{x}}{\mathrm{d}t}=\frac{\mathrm{d}x}{\mathrm{d}t}\hat{x}+x\times 0$$ Note that if you are not cartesian coordinates things aren't so simple, in spherical coordinates $$\frac{\mathrm{d}\hat{r}}{\mathrm{d}t}\neq 0$$ if the particle in question is changing coordinates.

Answer 2

Well $v_t$ here refers to the instantaneous speed of the subject particle and $a_t$ refers to instantaneous acceleration.

Let us look at this in 1D plane where x is the position vector of a small particle from the origin along the x axis.

We know very well that $$v = \frac{x}{t}$$where x is displacement in time t. ( Velocity is the rate of change of distance travelled in a particular direction)

Now , if we consider a small distance dx travelled in a small time dt by the particle ( along the x axis of course ), the velocity of the particle at a time t is ( Again, rate of change of displacement w.r.t time ) : $$v_t = \frac{d\vec{x}}{dt}$$ This is basically the instantaneous speed of the particle at a time t when x is a function of time t

Now since acceleration is the rate of change of velocity, the same applies to it as well.

This works for 2D and 3D as well by separately resolving the vectors and using this for each individual vector.

What Prof.Lewin has done is the same, except here he has written $\bar{r}$ ( which is called the position vector ) in terms of $\hat{x}$, $\hat{y}$ & $\hat{z}$ which are the position vectors along x axis, y axis and z axis respectively. When we differentiate this position vector we do it by individually differentiating in each direction as it is the sum of 3 functions basically.

If you still haven't understood the math, then you can probably find more about it under "Application of Derivatives", but in a math book, probably not in a physics book.