Explain the meaning of position, displacement, and distance traveled as they apply to an object moving along a line. Select the correct answer below. A. The position of an object moving along a line is the location of the object relative to some given starting position. The displacement is the distance from the origin at a given time \( t=a \). The distance traveled between \( t=a \) and \( t=b \) is \( \int_{a}^{b}|v(t)| d t \), where \( v(t) \) is the velocity at time \( t \). B. The position of an object moving along a line is the location of the object relative to the origin. The displacement is the change from initial position to time \( t=a \). The distance traveled between \( t=a \) and \( t=b \) is \( \int_{a}^{b} v(t) d t \), where \( v(t) \) is the velocity at time \( t \). C. The position of an object moving along a line is the location of the object relative to the origin. The distance traveled between \( t=a \) and \( t=b \) is \( \int_{a}^{b} v(t) d t \), where \( v(t) \) is the velocity at time \( t \). The displacement is the change in distance between time \( t=a \) and \( t=b \). D. The position of an object moving along a line is the location of the object relative to the origin. The displacement is the change in position between time \( \mathrm{t}=\mathrm{a} \) and \( \mathrm{t}=\mathrm{b} \). The distance traveled between \( \mathrm{t}=\mathrm{a} \) and \( \mathrm{t}=\mathrm{b} \) is \( \int_{\mathrm{a}}^{\mathrm{b}}|\mathrm{v}(\mathrm{t})| \mathrm{dt} \), where \( \mathrm{v}(\mathrm{t}) \) is the velocity at time t .

To understand the concepts of position, displacement, and distance traveled for an object moving along a line, let's break down each term: 1. **Position**: This refers to the specific location of an object at a given time relative to a reference point, often called the origin. It is typically represented as a coordinate on a number line. 2. **Displacement**: This is defined as the change in position of the object from one point in time to another. It is a vector quantity, meaning it has both magnitude and direction. Displacement can be calculated as the difference between the final position and the initial position. 3. **Distance Traveled**: This refers to the total length of the path taken by the object, regardless of direction. It is a scalar quantity, meaning it only has magnitude and no direction. The distance traveled can be calculated by integrating the absolute value of the velocity over a given time interval. Now, let's evaluate the provided options: - **Option A**: Incorrect because it misdefines displacement as the distance from the origin at a given time, which is not accurate. - **Option B**: Incorrect because it states that displacement is the change from initial position to time \( t=a \), which is not a complete definition. - **Option C**: Incorrect because it defines displacement as the change in distance, which is not accurate. Displacement is the change in position, not distance. - **Option D**: Correct. It accurately defines: - Position as the location relative to the origin. - Displacement as the change in position between times \( t=a \) and \( t=b \). - Distance traveled as \( \int_{a}^{b} |v(t)| dt \), which correctly accounts for the total distance regardless of direction. Thus, the correct answer is **D**.