Question

The initial position 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 initial position affects the intercept, not the slope.

Step-by-step solution

Understand the Position-Time Graph

A position-time graph depicts the position of an object (usually on the y-axis) as it varies with time (on the x-axis). If an object is moving with constant velocity, this graph is represented by a straight line.

Intercept and Slope Concept

In the context of the position-time graph, the *intercept* is the point where the line crosses the y-axis, which represents the position of the object at time zero. The *slope* of the line represents the velocity of the object.

Impact of Changing Initial Position

Increasing the initial position means you are changing the y-intercept because the object starts at a higher position on the y-axis at time zero. The slope, representing the velocity, remains unchanged since velocity is constant.

Conclusion

Increasing the initial position changes the y-intercept of the position-time graph while the slope remains constant, indicating that velocity is unchanged.

Key concepts

Constant Velocity

When an object moves at a constant velocity, its speed and direction stay the same over time. This means the object covers equal distances in equal intervals of time. In the context of a position-time graph, constant velocity is depicted as a straight line because the position of the object changes uniformly with time.

• A steeper slope indicates a higher velocity.
• A flatter slope shows a slower velocity.

A straight-line graph means there's no acceleration involved, just steady speed.

Y-Intercept

The y-intercept on a position-time graph represents the initial position of the object at the starting point, which is typically time zero. It is the point where the line crosses the y-axis.

• Changing the initial position of the object will result in a change in the y-intercept.
• The intercept value provides a clear snapshot of where the object begins its motion.

Increase the initial position, and you'll see the y-intercept shift upward. Lower it, and the intercept will move down.

Slope

The slope of a position-time graph is crucial because it represents the velocity of the object. A slope is essentially a measure of how steep the line is, which translates to how quickly the position is changing over time.

• Calculated as \( \text{slope} = \frac{\Delta y}{\Delta x} \) where \( \Delta y \) is the change in position, and \( \Delta x \) is the change in time.
• A positive slope means the object is moving forward.
• A negative slope indicates backward movement.

With a constant velocity, the slope remains unchanged even if the initial position is altered.

Initial Position

The initial position is where the object starts its journey in a position-time graph. This position determines the y-intercept of the graph but does not affect the slope.

• If you increase the initial position, it shifts the line up on the graph without changing its angle.
• If you decrease it, the line shifts lower.
• This parameter shows where exactly the object was at the beginning of observation.

The initial position establishes the starting point of the line, ensuring that changes don't impact velocity, hence the slope remains constant.