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Chapter 10: Problem 1

(a) By eliminating the parameter, sketch the trajectory over the time interval \(0 \leq t \leq 5\) of the particle whose parametric equations of motion are $$x=t-1, \quad y=t+1$$ (b) Indicate the direction of motion on your sketch. (c) Make a table of \(x\) - and \(y\) -coordinates of the particle at times \(t=0,1,2,3,4,5\) (d) Mark the position of the particle on the curve at the times in part (c), and label those positions with the values of \(t\)

Short Answer

Expert verified
The trajectory is a line from \((-1, 1)\) to \((4, 6)\), passing through specific points at each integer time from 0 to 5.

Step by step solution

01

Write the Parametric Equations

The equations given are: \( x = t - 1 \) and \( y = t + 1 \). These describe the motion of the particle in terms of the parameter \( t \).
02

Eliminate the Parameter

To eliminate the parameter \( t \), solve the equation for \( x \): \[ t = x + 1 \].Substitute this into the equation for \( y \) to get \( y = (x + 1) + 1 \), which simplifies to \( y = x + 2 \).
03

Sketch the Trajectory

The equation \( y = x + 2 \) describes a line with a slope of 1, passing through the point (0,2). Over the time interval \( 0 \leq t \leq 5 \), compute the corresponding \( x \) values (from \( t = x + 1 \)): - At \( t = 0 \), \( x = -1 \). - At \( t = 5 \), \( x = 4 \). The line segment goes from \( (-1, 1) \) to \( (4, 6) \).
04

Indicate Direction of Motion

Since \( x \) increases as \( t \) increases, the direction of motion is from the left \((-1, 1)\) towards the right \((4, 6)\) on the graph.
05

Create a Table of Coordinates

Calculate the \( x \) and \( y \) coordinates at given \( t \) values:- \( t = 0 \), \((x, y) = (-1, 1)\)- \( t = 1 \), \((x, y) = (0, 2)\)- \( t = 2 \), \((x, y) = (1, 3)\)- \( t = 3 \), \((x, y) = (2, 4)\)- \( t = 4 \), \((x, y) = (3, 5)\)- \( t = 5 \), \((x, y) = (4, 6)\)
06

Mark Positions on the Sketch

Place marks on the line for each position from Step 5, labeling them with the corresponding \( t \) values: - \((-1, 1)\) for \( t = 0 \)- \((0, 2)\) for \( t = 1 \)- \((1, 3)\) for \( t = 2 \)- \((2, 4)\) for \( t = 3 \)- \((3, 5)\) for \( t = 4 \)- \((4, 6)\) for \( t = 5 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trajectory Sketching
Sketching the trajectory of a particle using parametric equations involves visualizing the path of the particle over a given time interval. For the equations provided, where the motion of the particle is described by the equations
  • \( x = t - 1 \)
  • \( y = t + 1 \)
we first eliminate the parameter \( t \) to derive a single equation in terms of \( x \) and \( y \). Once eliminated, we have a more straightforward expression \( y = x + 2 \), which represents a line.

To sketch the trajectory, think of drawing this line on a graph, starting from the point where \( x = -1 \) and \( y = 1 \) (when \( t = 0 \)), and moving to \( x = 4 \) and \( y = 6 \) (when \( t = 5 \)). You can plot and then connect these points to visualize the line segment path of the particle. Remember that each point on the line corresponds to different times \( t \), showing the continuous movement of the particle from one point to another.
Eliminating the Parameter
The process of eliminating the parameter in a set of parametric equations is crucial for converting the particle's motion into a conventional function or equation. This allows us to express one variable purely in terms of the other, revealing the path's shape.

In our example, we start with two parametric equations:
  • \( x = t - 1 \)
  • \( y = t + 1 \)
By solving the first equation for \( t \), we get \( t = x + 1 \). Substituting \( t \) in the second equation, we replace \( t \) in \( y = t + 1 \) with \( x + 1 \), leading us to the equation:
\[ y = (x + 1) + 1 = x + 2 \]This is now a linear equation in standard form, making it much simpler to graph and understand. The line equation \( y = x + 2 \) visually represents how the two-dimensional motion unfolds on a Cartesian plane.
Coordinate Table Creation
Creating a coordinate table helps us systematically organize values of \( x \) and \( y \) at specific times \( t \). It also lays out how the particle moves across the plane, providing clear checkpoints along its path.

Given \( t \) values of 0 through 5, we compute the corresponding \( x \) and \( y \) values using the parametric equations:
  • For \( t = 0 \): \( x = -1 \), \( y = 1 \)
  • For \( t = 1 \): \( x = 0 \), \( y = 2 \)
  • For \( t = 2 \): \( x = 1 \), \( y = 3 \)
  • For \( t = 3 \): \( x = 2 \), \( y = 4 \)
  • For \( t = 4 \): \( x = 3 \), \( y = 5 \)
  • For \( t = 5 \): \( x = 4 \), \( y = 6 \)
These calculations create pairs like \((-1, 1)\) and \((4, 6)\), effectively marking the points on the trajectory at various times \( t \). This table is especially useful for freehand graphing or marking these points on a pre-drawn line to show the positions over time.

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