Vectors and Scalars — Question 3

Question 3

Different routes can have the same displacement vector

Two walkers start at A and finish at B, 6.0 m due east of A. Walker S follows the straight segment AB. Walker C follows the upper semicircle with AB as its diameter. Both trips take exactly 4.0 s, with no reversals along either stated route. Treat all dimensions as exact.

The scalar distance is total route length. The displacement vector points from the initial to the final position. Average velocity is displacement divided by elapsed time; average speed is distance divided by elapsed time. A semicircle of radius $r$ has arc length $\pi r$.

1. Find each displacement vector and traveled distance. Explain why the curvature of C’s route does not make its displacement a curved vector.

2. Find both average velocities and average speeds. Which quantities are equal between the walkers, and which are not?

3. Does either average determine a walker’s instantaneous speed at every time? Explain even for the straight route.

4. If C retraces the semicircle from B to A in a further 4.0 s, find displacement, distance, average velocity and average speed for C’s complete out-and-back trip. State how direction is handled for the zero displacement.

Question 3 - Solution

Different routes can have the same displacement vector

(a) The vector depends on endpoints; the scalar length depends on route.

$$\boxed{\Delta\vec r_S=\Delta\vec r_C=6\hat i\,\mathrm m},\quad s_S=6\,\mathrm m,\quad s_C=\pi(3)=3\pi\,\mathrm m$$ The semicircle radius is half the 6 m diameter. Displacement joins endpoints by a straight directed separation even when the traveled route curves.

(b) Divide each relevant quantity by the same duration.

$$\bar{\vec v}_S=\bar{\vec v}_C=\frac{6\hat i}4=1.5\hat i\,\mathrm{m/s},\quad\bar v_S=1.5,\quad\bar v_C=\frac{3\pi}4\quad\mathrm{m/s}$$ Both displacement vectors and average velocities agree. Route lengths and average speeds differ; the curved route has average speed greater than the magnitude of average velocity.

(c) An interval average does not specify timing along the route.

$$v_1=3/1=3,\quad v_2=3/3=1\quad\mathrm{m/s},\qquad\frac{v_1(1)+v_2(3)}4=1.5\,\mathrm{m/s}$$ These two stages cover 3 m each in 1 s and 3 s. They reproduce the straight walk’s average without constant speed. The curved walk likewise permits different timing along its fixed route.

(d) Returning cancels displacement, not accumulated distance.

$$\Delta\vec r=\vec0,\quad s=6\pi\,\mathrm m,\quad\bar{\vec v}=\vec0,\quad\bar v=\frac{6\pi}8=\frac{3\pi}4\,\mathrm{m/s}$$ Returning cancels directed separation but adds another semicircular arc. Zero displacement and zero average velocity have no defined direction; they do not imply no travel.