Question

The velocity of an object that moves with constant velocity is increased. Does this change the intercept or the slope of the position-time graph of the object's motion? Explain.

Short answer

Changing velocity affects the slope, not the intercept.

Step-by-step solution

Understanding the Problem

We are asked whether a change in velocity affects the slope or intercept of a position-time graph of an object's motion. The object initially moves with constant velocity and then its velocity increases.

Interpreting the Position-Time Graph

In a position-time graph, the slope represents the velocity of the object. The y-intercept represents the initial position when time is zero.

Analyzing the Impact of Velocity Change

Since the velocity changes, the slope of the position-time graph will be altered. With an increase in velocity, the slope becomes steeper. Since the y-intercept is dependent on the initial position, not the velocity, it remains unchanged.

Concluding the Effects

Thus, increasing the velocity changes the slope of the position-time graph while the intercept remains unchanged since the initial position is not altered.

Key concepts

Constant Velocity

In physics, when we talk about constant velocity, we mean that an object is moving at the same speed in a straight line over time. It does not speed up, slow down, or change direction. Constant velocity results in a straight line on a position-time graph, indicating that the object's position changes at a consistent rate.
This concept is important because it means that the object's motion can be easily predicted: just continue in that line’s path. On the graph, this consistent motion translates into a fixed slope, demonstrating a steady rate of change of position over time. As long as the velocity is constant, the graph's slope remains unchanged.

Slope and Intercept

When interpreting position-time graphs, understanding the slope and intercept is crucial. The slope of the graph represents the velocity of the object. In mathematical terms, the slope is defined as the change in position (y-axis) divided by the change in time (x-axis), which mirrors the velocity formula: \(v = \frac{\Delta x}{\Delta t}\).

• A steeper slope denotes a higher velocity.
• A flatter slope indicates a lower velocity.

The y-intercept, on the other hand, shows the starting point of the object on the graph. It is where the object is positioned at time zero. It doesn’t change with velocity adjustments because it’s tied to the object's initial position, not how fast it's moving.

Velocity Change

Changing an object's velocity means altering how quickly it moves, which directly affects the slope of the position-time graph. When velocity increases, the object covers more distance in the same amount of time, steepening the graph's slope. When velocity decreases, the slope becomes less steep, indicating slower movement.
However, it is important to note that while the slope will change with velocity variations, the y-intercept will not. This is because the intercept is linked to the object's position at the very start of the observation, and any change in velocity occurs after this starting point.
Therefore, in our context, a velocity change will only impact how steep the graph looks over time.

Graph Interpretation

Reading a position-time graph involves extracting information about an object's movement. The graph provides visual insight into how an object travels over time.
To interpret these graphs:

• Look at the slope to understand the velocity: steep slopes are fast, gentle slopes are slow.
• Note any changes in the slope as indicators of velocity changes.
• Check the y-intercept to find out where the journey starts on the graph.

A change in velocity can alter the slope, making interpretation of different graph segments essential. By understanding these aspects of the graph, one can draw conclusions about the object's motion throughout the time period observed.
Mastering the interpretation of these graphs enables students to translate visual data into meaningful motion insights.