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 the origin. The distance traveled between \( t= \) 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 \( t=b \). 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 in position between time \( t=a \) and \( t=b \). 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 some given starting position. The displacemen 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 \( t \). 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 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 \( t \).

The correct answer is: **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 in position between time \( t=a \) and \( t=b \). 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 \( t \).** ### Explanation: 1. **Position:** - **Definition:** The position of an object moving along a line is its location relative to a fixed reference point, typically the origin. - **Correct in Option B:** It accurately states that the position is relative to the origin. 2. **Displacement:** - **Definition:** Displacement is the change in position of the object. It is a vector quantity that considers both the magnitude and the direction from the initial position to the final position. - **Correct in Option B:** It correctly defines displacement as the change in position between two time instances, \( t=a \) and \( t=b \). 3. **Distance Traveled:** - **Definition:** Distance traveled is the total length of the path taken by the object, regardless of direction. It is a scalar quantity. - **Correct in Option B:** The distance traveled is given by the integral of the absolute value of velocity over the time interval, \( \int_{a}^{b}|v(t)| d t \), ensuring that all movement is accounted for positively, regardless of direction. ### Why Other Options Are Incorrect: - **Option A:** It incorrectly defines displacement as the change in distance rather than the change in position and does not take the absolute value of velocity for distance traveled. - **Option C:** It misdefines displacement as the distance from the origin at a given time rather than the change in position. - **Option D:** It incorrectly defines displacement and fails to use the absolute value of velocity for calculating distance traveled. Thus, **Option B** provides the most accurate and complete definitions of position, displacement, and distance traveled for an object moving along a line. **Answer:** **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 in position between time \( t=a \) and \( t=b \). 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 \( t \).**