Problem 5 - Calculus Early Transcendentals

Question

If a particle moves at constant velocity, what can you say about its position versus time curve?

Step-by-Step Solution

Answer

The position versus time curve is a straight line for constant velocity, indicating linear motion.

Step 1: Understand the Concept of Constant Velocity

When a particle moves at constant velocity, its speed and direction remain the same over time. This implies there are no changes or fluctuations in its motion.

Step 2: Recall the Equation of Motion for Constant Velocity

The position of a particle moving at constant velocity can be described by the equation \( x(t) = x_0 + vt \), where \( x(t) \) is the position at time \( t \), \( x_0 \) is the initial position, and \( v \) is the velocity.

Step 3: Describe the Shape of the Position-Time Graph

For constant velocity, the position \( x(t) \) changes linearly with time \( t \), since this relationship is linear: \( x(t) = x_0 + vt \). This results in a straight line on a position versus time graph.

Step 4: Interpret the Graph Slope

The slope of the position versus time graph represents the velocity. For a particle at constant velocity, the slope is constant, indicating a straight line graph, where the steepness of the line corresponds to the magnitude of the velocity.

Key Concepts

Equation of Motion

When a particle moves at constant velocity, it's important to understand the role of the Equation of Motion. Typically, when velocity is constant, the motion can be expressed using the equation: \( x(t) = x_0 + vt \). Here, \( x(t) \) represents the position of the particle at a specific time \( t \). Meanwhile, \( x_0 \) is the initial position when time \( t = 0 \), and \( v \) stands for the unchanged velocity.
This formula is quite straightforward, and it neatly summarizes how position relates to time when dealing with constant velocity. Each part of the equation plays an important role:

• \( x_0 \): The starting point, serving as the anchor where measurements begin from.
• \( vt \): This part shows how far the particle has traveled over time, based on its constant speed.

Understanding this equation helps in predicting the position at any given time, making it valuable in both analysis and solving motion-related problems.

Position-Time Graph

The position-time graph is a powerful tool for visualizing the motion of objects, especially when they travel with constant velocity. On this graph, time \( t \) is often set on the horizontal (x-axis) while position \( x(t) \) is plotted on the vertical (y-axis).
With constant velocity, this graph forms a straight line. The reason for this lies in the linear relationship between position and time described by the motion equation \( x(t) = x_0 + vt \). Each of these elements, time and position, determines the coordinates of successive points on the graph. As time progresses steadily, so does the position, in a way that maintains equal intervals - thus drawing a straight line.
This linear progression is a clear indicator of unchanging velocity, allowing easy interpretation of how quickly (or slowly) an object is moving. It helps students and professionals alike to assess motion without much calculation, by just observing the slope and direction of the line.

Linear Relationship

In physics and mathematics, a linear relationship is a direct proportionality between two variables. For constant velocity motion, the position \( x(t) \) and time \( t \) share such a relationship. This is because the position-time equation \( x(t) = x_0 + vt \) reveals a consistent rate of change.
The main features of a linear relationship include:

• **Proportional rate:** Position increases in direct proportion to time elapsed.
• **Consistency:** The change in position per unit of time remains constant.
• **Graph representation:** A straight, non-curved line on the graph.

This relationship simplifies understanding and predicting motion behavior, since it assures us that:

• No new patterns or changes emerge as time goes on.
• The graph doesn’t show any bends or angles, just straight lines.

For students, identifying linear relationships is crucial because it provides key insights into the concept of constant velocity and allows them to apply similar logical thinking to solve problems in other areas of study.