Question

Make a position-time graph for a particle that is at \(x=3.1 \mathrm{~m}\) at \(t=0\) and moves with a constant velocity of \(-2.7 \mathrm{~m} / \mathrm{s}\). Plot the motion for the range \(t=0\) to \(t=6.0 \mathrm{~s}\).

Short answer

Plot a straight line from (0, 3.1) to (6, -13.1) on a position-time graph.

Step-by-step solution

Understand the Problem

We are given the initial position of a particle at \( t=0 \), which is \( x=3.1 \text{ m} \). The particle moves with a constant velocity of \( -2.7 \text{ m/s} \). We need to create a position-time graph for the particle from \( t=0 \text{ s} \) to \( t=6.0 \text{ s} \).

Determine the Equation of Motion

The position \( x \) of a particle moving with constant velocity can be determined by the formula \( x = x_0 + vt \), where \( x_0 \) is the initial position, and \( v \) is the velocity. Here, \( x_0 = 3.1 \text{ m} \) and \( v = -2.7 \text{ m/s} \). Thus, the equation becomes \( x(t) = 3.1 - 2.7t \).

Calculate Position at Different Times

Using the equation \( x(t) = 3.1 - 2.7t \), calculate the position at various times: \( t=0 \text{ s}, 1.0 \text{ s}, 2.0 \text{ s}, ..., 6.0 \text{ s} \).

Set Up the Table of Values

Calculate the positions at different time points:- \( t=0.0 \text{ s} \), \( x(0) = 3.1 \text{ m} \)- \( t=1.0 \text{ s} \), \( x(1) = 3.1 - 2.7(1) = 0.4 \text{ m} \)- \( t=2.0 \text{ s} \), \( x(2) = 3.1 - 2.7(2) = -2.3 \text{ m} \)- Continue this until \( t=6.0 \text{ s} \) to calculate \( x(6) = -13.1 \text{ m} \).

Plot the Graph

Using the values from the table, plot a graph with time \( t \) on the horizontal axis and position \( x \) on the vertical axis. Mark the calculated points and draw a straight line through them, since the velocity is constant.

Key concepts

Understanding Constant Velocity

In the context of physics, constant velocity means that an object moves in a straight line at a steady pace, without accelerating or decelerating. This means:

• The speed does not change.
• The direction of motion remains the same.
• The distance covered is directly proportional to time.

Constant velocity is vital for plotting position-time graphs because if the velocity is constant, the graph will always be a straight line. This is particularly true because the slope of the line on a position-time graph represents the velocity. A flat line indicates zero velocity, whereas a sloped line indicates movement, with the steepness showing the velocity magnitude. This makes understanding and predicting motion straightforward when velocity is constant.

Equation of Motion

The equation of motion is essential for understanding how objects move over time, particularly when dealing with constant velocity. For an object moving with a constant velocity, the basic formula to calculate its position at any time is:\[x(t) = x_0 + vt\]Where:

• \(x(t)\) is the position at time \(t\).
• \(x_0\) is the initial position of the object.
• \(v\) represents constant velocity.

By plugging in the known values, like initial position and velocity, we can predict the future position of the object. This equation shows linearity, meaning any change in time results in a proportional change in position, a hallmark of constant velocity motion.

Graph Plotting Basics

Graph plotting is a graphical representation of an object's motion, making analysis more intuitive. With a position-time graph, we display how an object's position changes over time.

• The horizontal axis typically represents time \(t\), while the vertical axis shows position \(x\).
• Each point on the line indicates the position of the object at that specific time.

For our specific example, you'll calculate positions at several time intervals (e.g., \(t=0\) to \(t=6\) seconds) using the equation of motion, and plot these points accordingly. Drawing a line through these plotted points forms a straight line because of the constant velocity. This visual approach helps students understand relationships between time, velocity, and position.

Exploring Kinematics

Kinematics deals with the geometry of motion without considering the influencing forces, focusing instead on how objects move. Key aspects include understanding:

• Displacement, which is a vector quantity showing change in position.
• Velocity, measuring how fast and in what direction an object moves.
• Acceleration, describing changes in velocity over time (though not the focus here as we maintain constant velocity).

Through kinematics, we simplify complex motions into understandable frameworks using graphs and equations. By mastering kinematics, students not only understand how to analyze single motions like constant velocity, but also develop a foundation for more complex motion analysis involving varying velocities and accelerations.