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Chapter 6: Problem 71

Position of a Particle Suppose that the position of a particle moving along a straight line is given by $$s(t)=a t^{2}+b t+c$$ where \(t\) is time in seconds and \(a, b,\) and \(c\) are real numbers. If \(s(0)=5, s(1)=23,\) and \(s(2)=37,\) find the equation that defines \(s(t) .\) Then find \(s(8)\)

Short Answer

Expert verified
The equation is \(s(t) = -2t^2 + 20t + 5\) and \(s(8) = 37\).

Step by step solution

01

Substitute Initial Conditions into the Equation

The position of the particle is given by the equation \(s(t) = a t^2 + b t + c\). We have three conditions to work with: \(s(0) = 5\), \(s(1) = 23\), and \(s(2) = 37\). Using \(s(0) = 5\), substitute \(t = 0\) into the equation to find \(c\): \(s(0) = a(0)^2 + b(0) + c = 5\), so \(c = 5\).
02

Substitute and Create a System of Equations

Now that we know \(c = 5\), substitute the other conditions into the equation: For \(s(1) = 23\):\(a(1)^2 + b(1) + 5 = 23\).This simplifies to:\(a + b + 5 = 23\). (Equation 1)For \(s(2) = 37\):\(a(2)^2 + b(2) + 5 = 37\).This simplifies to:\(4a + 2b + 5 = 37\).(Equation 2)
03

Solve the System of Equations

Now let's solve the system of equations generated:Equation 1: \(a + b = 18\)Equation 2: \(4a + 2b = 32\)First, simplify equation 2:Divide the entire equation by 2:\(2a + b = 16\).Now, subtract Equation 1 from this result:\((2a + b) - (a + b) = 16 - 18\)The equation simplifies to:\(a = -2\).Substitute \(a = -2\) back into Equation 1:\(-2 + b = 18\).Solve for \(b\):\(b = 20\).
04

Write the Equation for s(t)

Substitute \(a\), \(b\), and \(c\) into the initial equation:\(s(t) = -2t^2 + 20t + 5\).
05

Evaluate s(t) at t = 8

To find \(s(8)\), substitute \(t = 8\) into the equation:\(s(8) = -2(8)^2 + 20(8) + 5\).Calculate each term:\(-2(64) = -128\),\(20(8) = 160\).Add the results:\(-128 + 160 + 5 = 37\).Thus, \(s(8) = 37\).

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Key Concepts

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

Particle Motion
When studying the motion of particles along a straight line, the core idea is to determine how their position changes over time. The position function, often denoted as \(s(t)\), provides a mathematical description of this motion.
In this particular exercise, the position of the particle is described by a quadratic expression \(s(t) = at^2 + bt + c\), where \(a\), \(b\), and \(c\) are constants that shape the trajectory based on initial conditions.
  • Time \(t\) is a variable that represents the specific moment we are interested in.
  • \(s(t)\) tells us exactly where the particle is along the line at any given time \(t\).
Understanding particle motion through this function is crucial, as it helps predict future positions and analyze a particle's path. It also forms the basis for solving real-world physics problems involving kinematic equations.
System of Equations
A system of equations is essentially a collection of two or more equations with multiple variables. The goal is to find a common solution that satisfies all the equations simultaneously. In the given exercise, we derived our system after substituting specific conditions into the position function.
  • Each equation in the system brings a new constraint.
  • We use these constraints to pinpoint the values of unknowns, like \(a\) and \(b\) in the quadratic function.
For our case, equations were derived by substituting \(t = 0\), \(t = 1\), and \(t = 2\) into the quadratic equation, eventually allowing us to solve for all constants required to define \(s(t)\) completely. Having these systems and knowing how to manipulate them is essential in algebra and calculus.
Substitution Method
The substitution method is a technique used to solve a system of equations by replacing one variable with an expression derived from another equation. Here鈥檚 a simplified approach to how it works in our example:
First, we determine one of the constants \(c\) directly from the condition \(s(0) = 5\). Now, with \(c\) known, the substitution method allows us to simplify the equations further to isolate and determine \(a\) and \(b\).
The steps follow this pattern:
  • Find one variable directly with given conditions.
  • Substitute this known value into other equations.
  • Simplify and isolate the remaining variables.
The benefit of substitution lies in its simplicity and effectiveness, especially when dealing with linear combinations of equations related to motion and similar mathematical models.
Position Function
The position function is a mathematical concept describing the location of a particle as a function of time, \(t\). In our exercise, the position function is quadratic: \(s(t) = a t^2 + b t + c\).
This equation gives a curve that predicts the motion of the particle and translates the relationship between position and time into mathematical terms.
  • The coefficient \(a\) effects acceleration. It defines how quickly the particle speeds up or slows down.
  • \(b\) affects the speed or rate at which the position changes over time.
  • \(c\) is the initial position, or where the particle begins at time zero.
Evaluating this function for different values of \(t\), like \(t = 8\), allows us to compute the position at specific times. Thus, understanding how to formulate and solve the position function is key in physics for predicting kinematic behaviors and analyses.

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Most popular questions from this chapter

A manufacturer of refrigerators must ship at least 100 refrigerators to its two West Coast warehouses. Each warehouse holds a maximum of 100 refrigerators. Warehouse A holds 25 refrigerators already, while warehouse \(B\) has 20 on hand. It costs \(\$ 12\) to ship a refrigerator to warehouse \(\mathrm{A}\) and \(\$ 10\) to ship one to warehouse B. How many refrigerators should be shipped to each warehouse to minimize cost? What is the minimum cost?

Draw a sketch of the two graphs described with the indicated number of points of intersection. (There may be more than one way to do this.) A circle and a parabola; four points.

Graph the solution set of each system of inequalities by hand. $$\begin{aligned} &e^{-x}-y \leq 1\\\ &x-2 y \geq 4 \end{aligned}$$

Solve each system graphically. Check your solutions. Do not use a calculator. $$\begin{aligned}&x^{2}+y^{2}=5\\\&x+y=3\end{aligned}$$

Concept Check Fill in each blank with the appropriate response. The graph of the system $$ y > x^{2}+2 $$ $$ \begin{array}{r} x^{2}+y^{2}<16 \\ y<7 \end{array} $$ consists of all points \(\frac{\text { the parabola given by }}{\text { (above/below) }}\) \(y=x^{2}+2, \frac{ }{\text { (inside/outside) }}\) the circle \(x^{2}+y^{2}=16,\) and (above/below) the line \(y =7\)
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