Vectors and Scalars — Question 7

Question 7

A speed-only sensor discards directional information

A point marker moves in a horizontal plane. At one instant a sensor reports speed exactly $5.0\ \mathrm{m/s}$ but no direction. A separate flag says only that the velocity has a strictly eastward part: directions due north, due south or anywhere westward are excluded. North and east are fixed laboratory directions.

For visualization, draw candidate velocity arrows with a common tail and a common scale. This diagram represents possible velocities at one instant, not possible positions of the marker.

1. Describe the full set of candidate arrows before using the flag. Describe the subset after the flag. Does the flag specify a unique velocity?

2. Give two distinct velocities compatible with both reports, and explain why storing only the speed cannot distinguish them.

3. What additional direction measurement would make the vector unique? Explain the role of its reference convention.

4. If a later exact speed reading is zero, how many velocity vectors are compatible? How should a simultaneous “strictly eastward” flag be interpreted then?

Question 7 - Solution

A speed-only sensor discards directional information

(a) Fix arrow length while allowing orientation to vary.

$$v_x^2+v_y^2=25\,\mathrm{m^2/s^2},\qquad\text{flag: }v_x>0$$ Candidate tips form a circle of radius 5 m/s. The flag restricts them to the open eastern semicircle, excluding north and south endpoints. Infinitely many directions remain.

(b) Distinct directions can share the same speed and coarse flag.

$$\boxed{\vec v_1=(5,0),\quad\vec v_2=(5\sqrt3/2,5/2)}\quad\mathrm{m/s},\qquad|\vec v_1|=|\vec v_2|=5\,\mathrm{m/s}$$ The two vectors satisfy both reports but differ in direction. A speed-only measurement maps them to the same number, losing information that cannot be recovered from that number alone.

(c) Supply one complete planar direction.

$$\vec v=5(\cos\theta\,\hat i+\sin\theta\,\hat j)\,\mathrm{m/s},\qquad\theta=30^\circ\ \Leftrightarrow\ \beta=60^\circ$$ A heading with a reference axis and turning sense uniquely fixes the planar direction. Magnitude and direction must refer to the same object, instant and frame.

(d) Zero magnitude collapses the family to one vector.

$$v=0\Rightarrow v_x^2+v_y^2=0\Rightarrow\vec v=\vec0,\qquad v_x=0\not>0$$ Only the zero velocity is compatible with zero speed. A simultaneous strictly eastward flag contradicts it; zero has undefined direction, not an arbitrary eastward direction.