Vectors and Scalars — Question 4

Question 4

A zero vector is complete without a direction

A marker travels one complete lap of a circular track of radius 2.0 m in 8.0 s, starting and finishing at the marked gate. It moves at constant nonzero speed during the lap. In a separate run, the marker stays at the gate for 8.0 s. All descriptions use the same laboratory reference.

A zero vector has magnitude zero and no defined direction. A nonzero velocity points in the instantaneous direction of motion, tangent to the circular route. Use circumference $2\pi r$.

1. Compare displacement, distance, average velocity and average speed in the two runs.

2. A file stores the lap displacement as “0 m, heading $270^\circ$.” Is the heading a physical direction of the zero vector? Give a clearer way to record it.

3. Are the lap’s instantaneous velocities zero because its average velocity is zero? Compare the velocities at the rightmost and leftmost points for counterclockwise travel.

4. Are two zero displacement vectors from different starting gates equal? Does that equality prove that their associated position records or paths are identical?

Question 4 - Solution

A zero vector is complete without a direction

(a) Distinguish a net change from accumulated travel.

$$\Delta\vec r_{\mathrm{lap}}=\Delta\vec r_{\mathrm{rest}}=\vec0,\quad s_{\mathrm{lap}}=4\pi\,\mathrm m,\quad\boxed{\bar v_{\mathrm{lap}}=\pi/2\,\mathrm{m/s}}$$ Both runs have zero average velocity. Rest has zero distance and speed. For the lap, the calculated average speed also equals instantaneous speed because constant speed is given.

(b) Zero magnitude supplies no orientation.

$$|\vec0|=0,\qquad\frac{\vec0}{|\vec0|}\text{ is undefined}$$ Record “0 m; direction undefined.” A stored heading of $270^\circ$ is merely a convention or stale field, not a direction measured for the zero vector.

(c) Instantaneous motion survives cancellation over a lap.

$$\vec v_{\mathrm{right}}=\tfrac\pi2\hat j\,\mathrm{m/s},\quad\vec v_{\mathrm{left}}=-\tfrac\pi2\hat j\,\mathrm{m/s},\quad\bar{\vec v}=\vec0$$ For counterclockwise travel, the rightmost tangent points north and the leftmost points south. The nonzero velocities cancel over the complete lap when integrated, giving zero net displacement.

(d) The zero vector is the same net change at every location.

$$\vec r_{f,1}-\vec r_{i,1}=\vec0=\vec r_{f,2}-\vec r_{i,2}\quad\not\Rightarrow\quad\vec r_{i,1}=\vec r_{i,2}$$ Zero displacements are equal regardless of their starting gates. They can arise from rest or from different closed routes; equality does not determine event locations or paths.