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Physics / Chapter 3: Vector / Section 3-1

Vectors and Scalars — Question 8

Section 3-1 · Physics pages 326–327

Question 8 Reversing an axis changes a signed description, not the motion Two observers use the same laboratory reference and metre scale on one east–west rail. Observer E declares east positive; observer W declares west positive. They record the same velocity of a cart at the same instant. E writes vE = −4.0 m/s. Both also measure the same temperature, −2.0 ◦C. E: positive east W: positive west same physical velocity: west Both coordinate arrows refer to the same rail; the lower arrow represents the cart velocity. (a) State the physical velocity direction and speed. What signed velocity value should W record? (b) A student says the observers disagree because their numbers have opposite signs. Explain what information must accompany each signed number before they can be compared. (c) Should W also reverse the temperature sign? Explain the difference between reversing a spatial axis and changing a temperature scale. (d) E later reports vE = +4.0 m/s for the cart. Compare this later physical vector with the initial vector. Contrast that actual change with merely changing observers’ axis conventions at one instant.
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Question 8 - Solution Reversing an axis changes a signed description, not the motion E: positive east W: positive west same physical velocity: west (a) Decode the sign using the declared positive direction. ˆeW = −ˆeE,⃗ v = (−4)ˆeE = (+4)ˆeW m/s, v = 4 m/s The cart physically travels west. Reversing the positive basis direction reverses the component sign and leaves the vector unchanged. (b) A signed value is meaningful only with its convention. vW = −vE, vW ˆeW = vEˆeE Compare signed numbers only after specifying axes, units, object, instant and frame. Opposite component signs here describe one physical velocity. (c) A spatial orientation is irrelevant to the scalar temperature. TW = TE = −2◦C Spatial axis reversal has no effect on a scalar temperature. Changing temperature units or the temperature zero would be a separate operation. (d) A new sign under a fixed convention denotes a new direction. ∆⃗v = [4 −(−4)]ˆeE = 8ˆeE m/s, vf = vi = 4 m/s Under the fixed E convention, the later measurement describes a physical reversal. Merely switching conventions at one instant produces no physical change. The two samples do not determine the intervening history.

Question 8

Reversing an axis changes a signed description, not the motion

Two observers use the same laboratory reference and metre scale on one east–west rail. Observer E declares east positive; observer W declares west positive. They record the same velocity of a cart at the same instant. E writes vE=−4.0m/sv_E=-4.0\ \mathrm{m/s}. Both also measure the same temperature, −2.0∘C-2.0\,{}^\circ\mathrm{C}.

Diagram shown in the original SVG worksheet.

  1. State the physical velocity direction and speed. What signed velocity value should W record?

  2. A student says the observers disagree because their numbers have opposite signs. Explain what information must accompany each signed number before they can be compared.

  3. Should W also reverse the temperature sign? Explain the difference between reversing a spatial axis and changing a temperature scale.

  4. E later reports vE=+4.0m/sv_E=+4.0\ \mathrm{m/s} for the cart. Compare this later physical vector with the initial vector. Contrast that actual change with merely changing observers’ axis conventions at one instant.

Question 8 - Solution

Reversing an axis changes a signed description, not the motion

Diagram shown in the original SVG worksheet.

(a) Decode the sign using the declared positive direction.

êW=−êE,v→=(−4)êE=(+4)êWm/s,v=4m/s\hat e_W=-\hat e_E,\quad\vec v=(-4)\hat e_E=(+4)\hat e_W\quad\mathrm{m/s},\quad\boxed{v=4\,\mathrm{m/s}} The cart physically travels west. Reversing the positive basis direction reverses the component sign and leaves the vector unchanged.

(b) A signed value is meaningful only with its convention.

vW=−vE,vWêW=vEêEv_W=-v_E,\qquad v_W\hat e_W=v_E\hat e_E Compare signed numbers only after specifying axes, units, object, instant and frame. Opposite component signs here describe one physical velocity.

(c) A spatial orientation is irrelevant to the scalar temperature.

TW=TE=−2∘CT_W=T_E=-2^\circ\mathrm C Spatial axis reversal has no effect on a scalar temperature. Changing temperature units or the temperature zero would be a separate operation.

(d) A new sign under a fixed convention denotes a new direction.

Δv→=[4−(−4)]êE=8êEm/s,vf=vi=4m/s\Delta\vec v=[4-(-4)]\hat e_E=8\hat e_E\,\mathrm{m/s},\qquad v_f=v_i=4\,\mathrm{m/s} Under the fixed E convention, the later measurement describes a physical reversal. Merely switching conventions at one instant produces no physical change. The two samples do not determine the intervening history.

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