Vectors and Scalars — Question 10

Question 10

Audit a record that mixes vectors, magnitudes and event data

A cart travels on a straight east-positive track over a 2.0 s interval. A reliable path-length recorder measures 10.0 m of total travel; a reliable comparison of endpoints establishes displacement 6.0 m east. The start position is not recorded. A software summary contains these proposed fields: The proposed average velocity is $5.0$ m/s east; the proposed average speed is $3.0$ m/s east. The proposed final instantaneous velocity is $3.0$ m/s east, and the proposed final position is $6.0$ m east of the laboratory origin. Treat the measured duration, distance and displacement as exact. Continuous position histories with ideal piecewise-constant velocities are permitted; no speed limit is supplied.

1. Correct the two average fields, including their quantity types. Explain why a direction field is inappropriate for average speed.

2. Construct one timed path matching the measurements and ending with westward velocity. Then construct one ending with eastward velocity. Give the stage distances and durations, and verify each path.

3. Assess the proposed final instantaneous velocity and final position. Which extra information would be needed to justify them?

4. Give a minimal corrected summary that separates measured interval facts, derived averages and still-undetermined quantities. Explain why attaching an arrow to every numerical field would not fix the record.

Question 10 - Solution

Audit a record that mixes vectors, magnitudes and event data

(a) Match each numerator to its physical definition.

$$\boxed{\bar{\vec v}=\frac{6\hat i}{2}=3\hat i\,\mathrm{m/s}},\qquad\bar v=\frac{10}{2}=5\,\mathrm{m/s}$$ Average velocity uses displacement; average speed uses total distance and carries no direction. The proposed average fields have exchanged magnitudes and improperly attached a direction to speed.

(b) Exhibit different final directions with the same interval data.

$$v_{\mathrm{west\ end}}=(+8,-2),\quad v_{\mathrm{east\ end}}=(-2,+8)\quad\mathrm{m/s}\text{ on successive 1-s stages}$$ Each pair gives displacement $8-2=6$ m, distance $8+2=10$ m and total time 2 s. Their final velocities are respectively westward and eastward. Ideal velocity jumps preserve continuous position as stipulated.

(c) Endpoint change and interval averages omit other information.

$$x_f=x_i+6\,\mathrm m,\qquad v(2^-)=-2\text{ or }+8\,\mathrm{m/s}\text{ in the two examples}$$ A final velocity measurement or sufficient motion model is needed to determine final velocity. A start coordinate or direct final coordinate is needed to locate the endpoint relative to the origin.

(d) Retain types and uncertainty rather than invented precision.

$$\Delta t=2\,\mathrm s,\quad s=10\,\mathrm m,\quad\Delta\vec r=6\hat i\,\mathrm m,\quad\bar v=5\,\mathrm{m/s},\quad\bar{\vec v}=3\hat i\,\mathrm{m/s}$$ Record the measured interval facts and derived averages above. Mark absolute endpoint positions and final instantaneous velocity undetermined. Arrow notation alone cannot repair an incorrect definition or supply missing event data.