Problem 79 - A Graphical Approach to Precalculus with Limits

Question

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 \(s(8)\)

Step-by-Step Solution

Answer

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

Step 1: Set Up Equations Using Given Conditions

We are given that the particle's position at different times leads to specific values. With the function \(s(t) = at^2 + bt + c\), we know:- \(s(0) = c = 5\)- \(s(1) = a(1)^2 + b(1) + c = 23\) - Substituting \(c = 5\), we get: \(a + b + 5 = 23\)- \(s(2) = a(2)^2 + b(2) + c = 37\) - With \(c = 5\), this becomes: \(4a + 2b + 5 = 37\).

Step 2: Solve for Constants a and b

Using the two equations derived from Step 1, we can solve for \(a\) and \(b\):- Simplifying \(a + b + 5 = 23\): \[a + b = 18\] - Simplifying \(4a + 2b + 5 = 37\): \[4a + 2b = 32\]Let's simplify the second equation by dividing the whole equation by 2: \[2a + b = 16\]Now solve this system of linear equations.

Step 3: Apply the Elimination Method

From Steps 1 and 2, we have two equations:1. \(a + b = 18\)2. \(2a + b = 16\)Subtract the first equation from the second:\[(2a + b) - (a + b) = 16 - 18\]\[a = -2\]Substitute \(a = -2\) back into the first equation:\[-2 + b = 18\]\[b = 20\].

Step 4: Determine the Position Function

We have found \(a = -2\), \(b = 20\), and \(c = 5\). Therefore, the position function \(s(t)\) is given by:\[s(t) = -2t^2 + 20t + 5\].

Step 5: Calculate \(s(8)\)

Substitute \(t = 8\) into the function \(s(t) = -2t^2 + 20t + 5\):\[s(8) = -2(8)^2 + 20(8) + 5\]Calculate each term:- \(-2(8)^2 = -2(64) = -128\)- \(20(8) = 160\)- Finally, add these into our equation:\[s(8) = -128 + 160 + 5\]\[s(8) = 37\].

Key Concepts

Particle Motion

Particle motion refers to the continuous movement of an object along a path. In our example, this path is a straight line. The movement is characterized by the position function \( s(t) \), which specifies the object's position at any given time \( t \). Particle motion is often depicted in various forms in physics, such as straight line or curvilinear motion. Typically, when working with particles, you plot their trajectory over time. Simplifying the equation governing the motion can help predict where the particle will be at future points in time. Understanding particle motion is essential for physics, where it connects concepts like speed, velocity, and acceleration for objects in motion.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form \( ax^2 + bx + c = 0 \). It is called 'quadratic' since it involves the square of the variable. In the context of our position function \( s(t) = at^2 + bt + c \), this equation beautifully models the trajectory of the particle as it moves. Quadratics are incredibly versatile because they can represent various motion scenarios, such as free-falling objects or projectiles.

• The variable \( t \) is raised to the power of 2.
• Coefficients \( a \), \( b \), and \( c \) determine the trajectory's shape.

Solving a quadratic equation often involves finding values for the variable that make the equation true, which means finding the roots of the equation. In this exercise, solving for \( a \), \( b \), and \( c \) used a system of equations derived from the given conditions.

System of Equations

A system of equations consists of multiple equations that are solved together to find common solutions. In our problem, we used them to determine the coefficients of the quadratic position function. By taking the conditions \( s(0) = 5 \), \( s(1) = 23 \), and \( s(2) = 37 \), we formed equations as follows:

• \( c = 5 \)
• \( a + b = 18 \)
• \( 2a + b = 16 \)

By solving this system, particularly through substitution and elimination methods, we found \( a = -2 \), \( b = 20 \), and confirmed \( c = 5 \). This process is vital in algebra, where you develop strategies to isolate variables and find their values systematically.

Time Variable

The time variable, usually denoted by \( t \), is fundamental in functions that describe motion. It serves as the input for the position function \( s(t) \), determining the particle's position at any point. As time progresses, the position changes, allowing us to track how fast and in what direction the particle moves. This concept is crucial since it links directly to the rate of change, leading to further exploration in derivatives and integrals.

• \( t \) is the independent variable in the function.
• It helps to analyze motion dynamics over specific intervals.

In our specific example, by plugging in \( t = 8 \), we calculated the particle's position using the function \( s(t) = -2t^2 + 20t + 5 \), resulting in \( s(8) = 37 \). Time is the key that unlocks understanding the sequences of movements in particle dynamics.