Question

Explain A straight line on a position-time graph represents motion with constant velocity. What does the slope of the line represent? What does the \(y\) intercept of the line represent?

Short answer

The slope of the line represents constant velocity; the y-intercept represents the initial position.

Step-by-step solution

Understanding the Position-Time Graph

Before analyzing the line, let's understand that a position-time graph shows how an object's position changes over time. A straight line indicates that the change in position over time is consistent, meaning the object is moving at a constant velocity.

Identifying the Slope

The slope of a line on a graph is calculated by the 'rise over run', which corresponds to the change in the vertical axis divided by the change in the horizontal axis. In this context, it represents the change in position (or distance) over the change in time, which is the object's velocity. Thus, the slope of the line represents the object's constant velocity.

Understanding the Y-Intercept

The vertical intercept, or the y-intercept, of a line on a graph occurs where the line crosses the y-axis. In the context of a position-time graph, this point represents the initial position of the object at time zero. Hence, the y-intercept indicates where the object started.

Key concepts

Constant Velocity

When we talk about constant velocity on a position-time graph, we're essentially describing a situation where an object moves with a steady speed in a straight line. There are no variations in how fast or slow the object moves; it's like a car cruising down the highway at an unchanging speed. On the graph, this steady motion is depicted as a straight line.
With constant velocity, the distance covered over each time unit is the same. This consistent travel is easy to see on the position-time graph because the line doesn't bend or curve.

• If the line is slanted upwards, the object is moving forward steadily.
• If the line is slanted downwards, the object is moving back steadily.
• If the line is horizontal, the object isn't moving at all. It stays in the same position as time progresses.

Understanding that a straight line equates to a constant velocity helps to visualize the motion, making it easier to grasp the concept of unchanged speed.

Slope of the Line

The slope of a line on a position-time graph holds a very special meaning. It essentially tells us about the object's velocity. The slope is calculated by examining how much the line moves vertically up or down for a certain movement along the horizontal axis.
To calculate the slope, follow these steps:

• Identify two points on the line.
• Calculate the vertical change (rise) between these two points.
• Calculate the horizontal change (run) between these two points.
• Divide the vertical change by the horizontal change: \[\text{slope} = \frac{\text{rise}}{\text{run}}\]

This calculated slope gives you the velocity of the object. If the slope is positive, the object moves forward; if it's negative, the object moves backward. A steeper slope indicates a faster speed.

• A slope of zero means the object is stationary, as there's no vertical movement over time.
• Understanding and calculating the slope is vital to interpreting these graphs.

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