Problem 2 - Calculus Early Transcendentals

Question

Suppose the velocity of an object moving along a line is positive. Are displacement and distance traveled equal? Explain.

Step-by-Step Solution

Answer

Answer: Yes, when an object has a positive velocity, the displacement and distance traveled will be equal. This is because the object moves in a single direction without changing its direction of motion, and the total length of the path traversed is the same as the change in position.

Step 1: Define Velocity

Velocity is a vector that describes the rate at which an object changes its position. A positive velocity indicates that an object is moving in the positive direction, and a negative velocity suggests that the object is moving in the negative direction. In this exercise, we are given that the object's velocity is positive, implying that it is moving in the positive direction along the line.

Step 2: Understand Displacement

Displacement is a vector quantity that represents the change in position of an object from its initial position to its final position. Mathematically, displacement is calculated as the difference between the final position (x_f) and the initial position (x_i) of an object: Displacement = x_f - x_i.

Step 3: Understand Distance Traveled

Distance traveled is a scalar quantity that represents the total length of the path traversed by the object, regardless of the direction of the movement. It measures how much ground an object has covered during its motion. Unlike displacement, distance traveled only has a magnitude and no direction.

Step 4: Comparing Displacement and Distance Traveled

When the velocity of an object is positive, it means the object is moving in the positive direction. In this case, the object will not change its direction of motion, so both the displacement and distance traveled will be in the same direction. As the object moves in the same direction, the difference between the initial and final positions will be the same as the total length of the path traversed by the object. So in this scenario, when the object has a positive velocity: Displacement = x_f - x_i Total Distance Traveled = |x_f - x_i| As the velocity is positive, the difference x_f - x_i will be positive. Thus, for an object with positive velocity, the displacement and distance traveled will be equal.

Step 5: Conclusion

When the velocity of an object moving along a line is positive, it means that the object is moving in a single direction. In this case, the displacement and distance traveled by the object will be equal. This is because the object does not change its direction, so the total length of the path traversed is the same as the change in position.