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Listed below are three pairs of initial and final positions that lie along the \(x\) axis. Using the concept of displacement discussed in Section \(2.1,\) decide which pairs give a positive displacement and which a negative displacement. Explain. $$ \begin{array}{|c|c|c|} \hline & \text { Initial position } x_{0} & \text { Final position } x \\ \hline(\mathrm{a}) & +2.0 \mathrm{~m} & +6.0 \mathrm{~m} \\ \hline(\mathrm{b}) & +6.0 \mathrm{~m} & +2.0 \mathrm{~m} \\ \hline(\mathrm{c}) & -3.0 \mathrm{~m} & +7.0 \mathrm{~m} \\ \hline \end{array} $$ Problem For each of the three pairs of positions listed above, determine the magnitude and direction (positive or negative) of the displacement. Verify that your answers are consistent with those determined in the Concept Question.

Step by step solution

01

Understanding Displacement

Displacement is defined as the change in position, which is the difference between the final position and the initial position along a straight line. It can be calculated using the formula \( \Delta x = x - x_0 \), where \( x \) is the final position and \( x_0 \) is the initial position. The direction of the displacement is determined by the sign of \( \Delta x \). A positive displacement means movement in the positive direction of the axis, while negative indicates movement in the opposite direction.

02

Calculate Displacement for Pair (a)

For pair (a), the initial position \( x_0 = +2.0 \text{ m} \) and the final position \( x = +6.0 \text{ m} \). Calculate the displacement: \[ \Delta x = x - x_0 = 6.0 \text{ m} - 2.0 \text{ m} = +4.0 \text{ m} \]. The displacement is positive, indicating movement in the positive direction along the \( x \)-axis.

03

Calculate Displacement for Pair (b)

For pair (b), the initial position \( x_0 = +6.0 \text{ m} \) and the final position \( x = +2.0 \text{ m} \). Calculate the displacement: \[ \Delta x = x - x_0 = 2.0 \text{ m} - 6.0 \text{ m} = -4.0 \text{ m} \]. The displacement is negative, indicating movement in the negative direction along the \( x \)-axis.

04

Calculate Displacement for Pair (c)

For pair (c), the initial position \( x_0 = -3.0 \text{ m} \) and the final position \( x = +7.0 \text{ m} \). Calculate the displacement: \[ \Delta x = x - x_0 = 7.0 \text{ m} - (-3.0 \text{ m}) = 10.0 \text{ m} \]. The displacement is positive, indicating movement in the positive direction along the \( x \)-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Initial Position

The initial position is the starting point of our journey along the axis. It's where everything begins before any movement takes place. In the world of physics, this is often denoted by the symbol \(x_0\). Consider it as the coordinate that gives you a reference or a marker from which any subsequent movement is measured.

• For pair (a), the initial position is \(+2.0\, \text{m}\).
• In pair (b), it's \(+6.0\, \text{m}\).
• Similarly, for pair (c), the initial position is \(-3.0\, \text{m}\).

The initial position helps us determine how far something has traveled and in what direction it's headed or been displaced compared to where it started. Understanding this concept ensures a clear grasp of how displacement transforms from just a mere distance to a meaningful directional movement.

Final Position

The final position is where the journey ends. It's the last known location along the axis after the movement. In formulaic terms, it's labeled as \(x\). Knowing the final position helps in calculating how much and in which direction an object has moved from its origin.

• For pair (a), the final destination is \(+6.0\, \text{m}\).
• In pair (b), the journey concludes at \(+2.0\, \text{m}\).
• Lastly, for pair (c), the final location is \(+7.0\, \text{m}\).

By comparing the final position with the initial position, you can determine the displacement, giving you insight into not only the magnitude of travel but also the specific direction along the axis.

Direction of Displacement

The direction of displacement tells us which way on the axis the movement has occurred. Displacement is not merely a measure of distance but also inherently involves direction, which can be identified as positive or negative.

• A positive displacement means moving towards higher numerical values on the axis (to the right).
• Negative displacement indicates movement towards lower numerical values (to the left).

For pair (a), the calculated displacement is \(+4.0\, \text{m}\), signifying a positive direction.
For pair (b), the displacement is \(-4.0\, \text{m}\), indicating a negative direction.
Finally, for pair (c), the displacement measures \(+10.0\, \text{m}\), clearly pointing in a positive direction.Understanding direction is crucial because it helps you visualize real-world scenarios of the object's path and movement along the one-dimensional axis.