Problem 2 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Answer: No, the position, displacement, and distance traveled aren't always equal. However, in a scenario with constant positive velocity, the object's displacement and distance traveled can be equal since the object is moving only in a positive direction along the line. The position is increasing with time but is not equal to displacement or distance traveled.

Step 1: Understand the definitions of position, displacement, and distance traveled

Position refers to the location of an object in a specified coordinate system. Displacement is the change in position of an object, which is the difference between its final position and initial position. Distance traveled is the total length of the path covered by the object.

Step 2: Examine the given condition - positive velocity

In this problem, we are given that the velocity of an object is positive. Velocity is the rate at which an object changes its position with time. A positive velocity indicates that the object is moving in a positive direction along the line.

Step 3: Analyze the relationship between position, displacement, and distance traveled

Since the velocity is positive, it implies that the object is moving from a low position value to a higher one, which means its position is increasing. Displacement is the difference between the final and initial positions, which would also be positive since the object is moving in a positive direction. Distance traveled is always a non-negative value, as it represents the path covered by the object, ignoring the direction. In this case, since the object is only moving in a positive direction, its distance traveled would be equal to the displacement.

Step 4: Conclusion

In a scenario where the velocity of an object is positive, the position, displacement, and distance traveled aren't always equal. However, their relationship would be: 1. The object's position is increasing with time. 2. The object's displacement is positive and equal to the difference between its final and initial positions. 3. The object's distance traveled is equal to its displacement, as it is moving only in a positive direction along the line. It is important to note that this scenario assumes a constant positive velocity, and the equality between displacement and distance traveled may change if the object's velocity becomes negative or zero at any point during its motion.

Key Concepts

Position

Position refers to the specific location of an object within a given coordinate system. Imagine plotting this as a point on a number line or in a plane. It tells us **where** the object is at any given moment.

When we speak about the position, several factors can influence this, such as time, movement, and sometimes even the environment the object is in. But in essence, position is all about locating a point in space relative to a known reference point.

• Position is often represented numerically in units like meters or feet.
• It's a snapshot of location at a specific time.
• Changes in position help us determine movement or motion.

Whenever velocity is discussed, position becomes integral because, as the object moves, its position changes. When velocity is positive, the position increments in the direction defined as positive within the coordinate system. So, in problems like the one described, the object's position continually increases. This increasing position is what leads to an understanding of the concepts like displacement and distance traveled.

Displacement

Displacement is all about how far an object has moved from its starting point to its endpoint. It is a vector quantity, meaning it has both magnitude and direction.

To calculate displacement, you simply take the final position and subtract the initial position, then consider the direction in which the object moved.

• Displacement can be positive, negative, or even zero.
• It solely depends on the start and end points, not the path taken in between.
• Measured in straight-line distance between two points.

This makes displacement very different from simply tracking the path of motion. In the context of a positive velocity, like in the example, the displacement is positive because the object continues moving in the same direction from start to end point. Importantly, while position provides a static point, displacement gives insight into the motion relative to another point, which is a crucial aspect when discussing motion and velocity.

Distance Traveled

Distance traveled tells about the ground an object has covered over time, completely ignoring the direction. It's a scalar quantity, indicating it only has magnitude. Whether the object circled around or zig-zagged back and forth, distance accounts for every step taken.

The key differences when considering distance traveled are:

• Always positive, as it measures the total travel regardless of the path.
• Measured by summing every step along the path taken.
• Not concerned with the endpoint's direct distance from the start point.

In a scenario with a constant positive velocity, as described in the problem, the distance traveled will match the displacement. This is simply because there are no changes in direction; the path from start to finish is straightforward. Understanding this highlights that while distance traveled gives a complete movement picture, it does not address how far the starting point is from the ending point in terms of direct straight-line measurement, which is exactly what the displacement concept covers.