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Chapter 3: Problem 76

Demonstrate with a vector diagram that a displacement is the same when measured in two different reference frames that are at rest with respect to each other.

Short Answer

Expert verified
Explain your answer. Answer: Yes, the displacement of an object remains the same when measured in two different reference frames that are at rest with respect to each other. This is because both frames have the same relative motion, and when we compare the displacement vectors in both reference frames, they have the same magnitude and direction.

Step by step solution

01

Define the displacement vector in the first reference frame

Let's consider an object that moves from point A to point B, and we have two reference frames R1 and R2, both at rest with respect to each other. We can represent the displacement of the object in reference frame R1 as a vector \(\vec{D1}\).
02

Define the displacement vector in the second reference frame

Similarly, we can represent the displacement of the object in reference frame R2 as a vector \(\vec{D2}\).
03

Create the vector diagram for R1

Now, let's create a vector diagram for reference frame R1. Draw a coordinate system with origin O1 and plot points A and B in this coordinate system, representing the initial and final positions of the object, respectively. Connect points A and B with an arrow, representing vector \(\vec{D1}\).
04

Create the vector diagram for R2

Create a vector diagram for reference frame R2. Draw a coordinate system with origin O2, and plot points A and B in this coordinate system, representing the initial and final positions of the object, respectively. Connect points A and B with an arrow, representing vector \(\vec{D2}\).
05

Compare the displacement vectors

Since both reference frames R1 and R2 are at rest relative to each other, the displacement vectors \(\vec{D1}\) and \(\vec{D2}\) should have the same magnitude and direction. Comparing the vector diagrams, we can see that the displacement vectors have the same length and are parallel. Hence, the displacement of the object is the same when measured in two different reference frames that are at rest with respect to each other.

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Most popular questions from this chapter

A small plane is flying directly west with an airspeed of $30.0 \mathrm{m} / \mathrm{s} .\( The plane flies into a region where the wind is blowing at \)10.0 \mathrm{m} / \mathrm{s}\( at an angle of \)30^{\circ}$ to the south of west. In that region, the pilot changes the directional heading to maintain her due west heading. (a) What is the change she makes in the directional heading to compensate for the wind? (b) After the heading change, what is the ground speed of the airplane?
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Geoffrey drives from his home town due east at \(90.0 \mathrm{km} / \mathrm{h}\) for 80.0 min. After visiting a friend for 15.0 min, he drives in a direction \(30.0^{\circ}\) south of west at \(76.0 \mathrm{km} / \mathrm{h}\) for 45.0 min to visit another friend. (a) How far is it to his home from the second town? (b) If it takes him 45.0 min to drive directly home, what is his average velocity on the third leg of the trip? (c) What is his average velocity during the first two legs of his trip? (d) What is his average velocity over the entire trip? (e) What is his average speed during the entire trip if he spent 55.0 min visiting the second friend?
In each of these, the \(x\) - and \(y\) -components of a vector are given. Find the magnitude and direction of the vector. (a) $A_{x}=-5.0 \mathrm{m} / \mathrm{s}, A_{y}=+8.0 \mathrm{m} / \mathrm{s} .\( (b) \)B_{x}=+120 \mathrm{m}$ $B_{y}=-60.0 \mathrm{m} .(\mathrm{c}) C_{x}=-13.7 \mathrm{m} / \mathrm{s}, C_{y}=-8.8 \mathrm{m} / \mathrm{s} .(\mathrm{d}) D_{x}=$ \(2.3 \mathrm{m} / \mathrm{s}^{2}, D_{y}=6.5 \mathrm{cm} / \mathrm{s}^{2}\)
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