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Chapter 10: Problem 49

A particle moves along the curve described by the parametric equations \(x=f(t), y=g(t) .\) Use a graphing utility to draw the path of the particle and describe the notion of the particle as it moves along the curve. $$x=2 t, \quad y=4 t-t^{2} \quad 0 \leq t \leq 6$$

Short Answer

Expert verified
The particle moves along the curve described by the parametric equations \(x=2t\), \(y=4t-t^{2}\), \(0 \leq t \leq 6\). It starts at the origin (0,0) and moves along the curve in the positive x direction. It reaches its highest point at (8,8) when \(t=4\), and then follows the curve back down in the negative y direction until it reaches the point (12, 0) when \(t=6\).

Step by step solution

01

1. Identify the parametric equations given.

The parametric equations given are: \(x=2t\), \(y=4t-t^{2}\), with \( 0 \leq t \leq 6 \) From these equations, you can observe that \(x\) and \(y\) are functions of the parameter \(t\).
02

2. Remove the parameter to find the corresponding Cartesian equation.

You can replace \(x\) with the equation given in terms of \(t\), resulting in: \(x = 2t \Longrightarrow t = \frac{x}{2}\) Now you can replace \(t\) in the \(y\) equation with this expression: \(y = 4\left(\frac{x}{2}\right) - \left(\frac{x}{2}\right)^{2} = 2x - \frac{x^2}{4}\)
03

3. Plot the graph of the Cartesian equation using a graphing utility.

Using a graphing utility, plot the graph of the equation \(y = 2x - \frac{x^2}{4}\) with \(\) and keeping in mind the constraints \( 0 \leq t \leq 6\) and \(0 \leq x \leq 12\), since \(x = 2t\).
04

4. Describe the motion of the particle along the curve.

As the parameter \(t\) increases from 0 to 6, the values of \(x\) also increase from 0 to 12. The particle moves along the curve as follows: - Initially, at \(t = 0\), the particle starts at the origin, \((0,0)\). - As \(t\) increases, the particle moves along the curve in the positive x direction. The particle reaches the highest point on the curve when the value of \(t\) is 4. At this point, the coordinates are \(\left(8, 8\right)\). - As \(t\) increases further, the particle continues moving along the curve in the positive x direction but now starts moving in the negative y direction until it reaches the point \(\left(12, 0\right)\) when \(t = 6\). Thus, by analyzing the graph, the motion of the particle is described as following the curve from the origin, reaching the highest point at (8,8), and then following the curve back down to the x-axis when t = 6.

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Key Concepts

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

Cartesian Equation
When dealing with parametric equations, a common step is converting them into a single equation known as the Cartesian equation. This eliminates the parameter and expresses the relationship directly between the two variables, typically denoted as x and y.
For example, in our exercise, the parametric equations given are:
  • \(x = 2t\)
  • \(y = 4t - t^2\)
To find the Cartesian equation, solve the x-equation for the parameter \(t\). From \(x = 2t\), get \(t = \frac{x}{2}\).
Use this expression to replace \(t\) in the y-equation:
  • \(y = 4\left(\frac{x}{2}\right) - \left(\frac{x}{2}\right)^2\)
Simplifying gives the Cartesian equation: \(y = 2x - \frac{x^2}{4}\).
This equation represents a parabola that can now be analyzed within the specified range, helping us study the motion without involving the parameter \(t\).
Graphing Utilities
Graphing utilities are powerful tools that can visualize mathematical equations and their graphs with ease. They are especially useful for parametric equations since they can plot both x and y in terms of the parameter \(t\).
In this particular problem, to graph the motion of the particle described by the Cartesian equation \(y = 2x - \frac{x^2}{4}\), a graphing utility helps in several ways:
  • It plots the curve for the specified range, which in this case is for \(0 \leq x \leq 12\) as translated from \(0 \leq t \leq 6\).
  • This visualization allows for a more intuitive understanding of the path and behaviour of the particle as it moves along the curve.
  • It makes it easier to see important points such as where the graph starts, its peak, and where it ends.
With graphing utilities, identifying these features and traits on the graph becomes simpler, drastically aiding in comprehension of the particle's dynamics.
Particle Motion Analysis
Analyzing particle motion using parametric equations offers insights into how the particle behaves over time within a defined range.
In our exercise:
  • At \(t = 0\), the particle begins at the origin \((0,0)\), indicating its starting point.
  • As \(t\) increases, both x and y change, revealing how the particle shifts positions along its path.
At a certain point, \(t = 4\), the particle reaches its maximum height at \((8,8)\). It shows that the particle isn't just moving along a straight line but rather follows a curved path.
  • After this peak, the particle starts descending until \(t = 6\) where it returns to the x-axis at \((12,0)\).
This behavior describes a parabolic motion typical when both vertical and horizontal movement components are involved. Hence, this helps in comprehending how factors like time affect position and how they translate graphically.

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Most popular questions from this chapter

Find the a:ea of the surface generated by revolving the curve about the \(x\) -axis. \(y=\cos x . x \in\left[0, \frac{1}{2} \pi\right]\).

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