Chapter 2: Problem 83
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
Expert verified
Plot a straight line from (0, 3.1) to (6, -13.1) on a position-time graph.
Step by step solution
01
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 \( t=6.0 \text{ s} \).
02
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 \( x(t) = 3.1 - 2.7t \).
03
Calculate Position at Different Times
Using the equation \( x(t) = 3.1 - 2.7t \), calculate the position at various times: \( t=0 \text{ s}, 1.0 \text{ s}, 2.0 \text{ s}, ..., 6.0 \text{ s} \).
04
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 \( x(6) = -13.1 \text{ m} \).
05
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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.
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).