Question

Gon 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 represents velocity, and the y-intercept represents the initial position.

Step-by-step solution

Understand Position-Time Graph

In a position-time graph, the vertical axis (y-axis) represents the position of an object, while the horizontal axis (x-axis) represents time. The graph shows how the position of the object changes over time.

Interpret the Slope

The slope of a line on a position-time graph is calculated by the formula \( abla = \frac{\Delta y}{\Delta x} \), where \( \Delta y \) is the change in position and \( \Delta x \) is the change in time. This slope represents the velocity, which is the rate of change of position with respect to time. A constant slope indicates constant velocity.

Analyze the Y-Intercept

The y-intercept of a position-time graph is the point where the line crosses the y-axis, which corresponds to the starting position of the object when time, \( t = 0 \). It indicates the initial position of the object when the observation or recording began.

Key concepts

constant velocity

Constant velocity on a position-time graph is represented by a straight line. This is because the position of the object changes uniformly with time, meaning that the speed and direction remain the same throughout the motion. A constant velocity indicates that there is no acceleration involved, as the object's motion does not change over time.
This constant nature simplifies the motion, making it easy to predict the future position of the object.

• The key feature of constant velocity is its unchanging nature.
• It implies equal distances are covered in equal intervals of time.
• In mathematical terms, velocity is constant if the slope is the same at any point on the graph.

This leads us to understand the concept of the slope of the line that indicates the velocity.

slope of the line

The slope of a line on a position-time graph is crucial as it determines the velocity of the object in motion. Mathematically, the slope is calculated by the formula \( abla = \frac{\Delta y}{\Delta x} \), where \( \Delta y \) represents the change in position and \( \Delta x \) represents the change in time.
In simple terms, it tells us how fast the position of the object is changing over time. A steep slope indicates a high velocity, while a gentle slope indicates a lower velocity.

• The slope can be positive, negative, or zero.
• A positive slope means the object is moving forward.
• A negative slope indicates reverse motion.
• Zero slope suggests the object is at rest.

Understanding the slope helps in characterizing the motion effectively.

y-intercept

The y-intercept on a position-time graph is quite significant as it provides the initial condition of motion. It is the point where the line crosses the y-axis, which corresponds to the object's position when time \( t = 0 \).
This position is referred to as the initial position and indicates where the object started from at the beginning of observation or measurement.

• The y-intercept is foundational for understanding the starting point of motion.
• It helps in drawing the complete picture of motion over time.
• Knowing the y-intercept allows us to predict the position of the object as time progresses.

The y-intercept acts as the anchor point, from which the object's journey can be traced.

velocity

Velocity is a fundamental concept in understanding motion on a position-time graph. Unlike speed, velocity includes both the magnitude and direction of motion, making it a vector quantity.
In the context of a position-time graph, the slope represents the velocity. This means the velocity is the rate at which an object’s position changes over time.

• A straight horizontal line indicates zero velocity.
• Positive slope signifies motion in the positive direction.
• Negative slope suggests motion in the negative direction.

Understanding velocity helps in predicting the motion's outcome, such as where the object will be after a given time period.

initial position

The initial position of an object on a position-time graph is depicted by the y-intercept. It marks the starting position of the object at the beginning of the observation period when time is zero.
This information allows us to trace back the steps of the object's journey and understand its travel path.

• Initial position provides insight into the motion's starting condition.
• It forms the baseline for calculating changes in position over time.
• Knowledge of the initial position aids in forming complete equations of motion.

An accurate understanding of the initial position is key to solving more complex motion problems.