17  Problems 1

I’ll use this Section to publish the answers to student questions. I’ll keep the question askers anonymous.

17.1 Units of the constants \(a\), \(b\) and \(c\)

In Section 9.4 part (a), why do the units of \(a\), \(b\) and \(c\) each include \(\text{m}^{-1}\)? I thought the point of these constants was to cancel the units of time (seconds) and leave units of length (metres) on the left-hand side. Putting \(\text{m}^{-1}\) on the right-hand side seems to mismatch the powers of metres: the left-hand side has \(\text{m}^{1}\), but the constants look like they have \(\text{m}^{-1}\).

The \(\text{m}^{-1}\) belongs to the constants alone, not to each whole term in \(\vec{r}\). The constants sit in the denominator. For example, the term \(t^2/b\) must have units of metres, so \[ [b] = \frac{[t]^2}{[\vec{r}]} = \frac{\text{s}^2}{\text{m}} = \text{m}^{-1}\text{s}^2. \] Dividing by something with units \(\text{m}^{-1}\) is the same as multiplying by metres, which is how the length units appear on the left-hand side. The same idea gives \([c] = \text{m}^{-1}\text{s}^3\) and \([a] = \text{m}^{-1}\text{s}\).

17.2 Moving parallel to \(\hat{\imath}\)

In Section 9.4 part (b), why can I not assume that when the particle is moving parallel to \(\hat{\imath}\), the \(\hat{\jmath}\) component of its position is zero? Why does it have to be the \(\hat{\jmath}\) component of the velocity?

“Moving parallel to \(\hat{\imath}\)” refers to the direction of motion, which is the direction of the velocity \(\vec{v}\), not the position \(\vec{r}\). A particle can be at a position with a non-zero \(\hat{\jmath}\) component (somewhere “above” the \(x\)-axis) and still be travelling purely in the \(\hat{\imath}\) direction. Setting the \(\hat{\jmath}\) component of \(\vec{r}\) to zero would instead mean the particle is on the \(x\)-axis, which is a different condition.

17.3 Galilean transformation \(k\)-component

For the last question in problems class 1 (Section 9.6), when I worked through it my \(k\) component was around 55, however in the notes it says 58 (my \(i\) and \(j\) components were the same as the ones in the notes once rounded). Did I go wrong somewhere?

The bird has no vertical velocity so the vertical distance from the bird to the ground is always \(58\text{ m}\). This means that the \(\hat{k}\) component of the displacement, in the bird’s frame, when the book hits the ground is \(-58 \hat{k}\text{ m}\).

Perhaps you made a sign error and subtracted 2 instead of adding 2?

17.4 Setting up vector diagrams for Galilean transformations

I struggled to set up the diagram for Section 9.6 and was wondering if you had any advice, practice questions, or videos for setting up vector diagrams and practising Galilean transformations with vectors?

Draw the positions at a general time \(t\), not just \(t = 0\): Galileo at the origin, the bird at \(\vec{r}_0 + \vec{v}t\), and the book at \(\vec{r}(t)\). The book’s position in the bird’s frame is the vector from the bird to the book, \(\vec{r'}(t) = \vec{r}(t) - (\vec{r}_0 + \vec{v}t)\). See Figure 9.1 and Figure 9.2, and the 1D version in Figure 4.1.

For more practice, try the relative-velocity and frames examples in the course textbook “Physics for scientists and engineers” by Tipler & Mosca which is available in the Physics Library and on-line version via the Library website. You can read more about relative velocity in §3.1 of the textbook which includes some wokred examples. There are more questions at the end of the chapter that you can try, with answers to odd numbered questions are at the back to the textbook.