Problem 72 - A Graphical Approach to College Algebra

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 \(s(0)=-10, s(1)=6,\) and \(s(2)=30,\) find the equation that defines \(s(t) .\) Then find \(s(10)\)

Step-by-Step Solution

Answer

The equation is \(s(t) = 4t^2 + 12t - 10\), and \(s(10) = 510\).

Step 1: Plug in Initial Conditions

To find the values of \(a\), \(b\), and \(c\), we substitute the given conditions into the equation \(s(t) = a t^2 + b t + c\). Starting with \(s(0) = -10\), plug in \(t = 0\): \[-10 = a(0)^2 + b(0) + c\]. This simplifies to \(c = -10\).

Step 2: Substitute into Equation for s(1)

Now, use \(s(1) = 6\):\[6 = a(1)^2 + b(1) + (-10)\]. Which simplifies to \[6 = a + b - 10\]. Therefore, \[a + b = 16\].

Step 3: Substitute into Equation for s(2)

Next, use \(s(2) = 30\): \[30 = a(2)^2 + b(2) + (-10)\]. This simplifies to \[30 = 4a + 2b - 10\]. Thus, \[4a + 2b = 40\].

Step 4: Solve System of Equations

Now solve the system of equations:1. \(a + b = 16\) 2. \(4a + 2b = 40\)Multiply the first equation by 2 to align with the second:\[2a + 2b = 32\].Subtract it from the second equation:\[(4a + 2b) - (2a + 2b) = 40 - 32\]\[2a = 8\]\[a = 4\].Substitute \(a = 4\) back into \(a + b = 16\):\[4 + b = 16\]\[b = 12\].

Step 5: Write Final Equation of Motion

Now that we know \(a = 4\), \(b = 12\), and \(c = -10\), substitute these back into the original formula:\[s(t) = 4t^2 + 12t - 10\].

Step 6: Calculate s(10)

To find \(s(10)\), substitute \(t = 10\) into the equation:\[s(10) = 4(10)^2 + 12(10) - 10\].Calculate:\[s(10) = 4(100) + 120 - 10 = 400 + 120 - 10 = 510\].

Key Concepts

Position of a Particle

In physics, the position of a particle refers to its location along a straight path. Here, the position is described by a quadratic function of time, given by \(s(t) = a t^2 + b t + c\). This equation models the motion of the particle as time \(t\) changes. The coefficients \(a\), \(b\), and \(c\) determine the specific path the particle takes. More specifically:

• \(a\) affects how the particle accelerates over time, causing the path to curve.
• \(b\) influences the velocity or speed at which the particle moves.
• \(c\) is the initial position of the particle when \(t = 0\).

By plugging in known conditions, such as \(s(0)\), \(s(1)\), and \(s(2)\), we can solve for these coefficients to describe the particle's motion accurately. This enables us to predict its position at any time \(t\), such as \(s(10)\).

Time-Dependent Equations

Time-dependent equations are any equations in which the variables change with time. In this scenario, we use the quadratic equation to express how the position of a particle changes as time progresses. Quadratic functions are especially useful in modeling situations with curved paths, like the motion of particles, because they provide a simple yet effective way to describe acceleration. In this particular problem, the equation \(s(t) = 4t^2 + 12t - 10\) is derived from given conditions. By substituting \(s(0) = -10\), \(s(1) = 6\), and \(s(2) = 30\) into the equation, we find the values for \(a\), \(b\), and \(c\), giving us a complete picture of the particle's path over time. These coefficients precisely define how quickly and in what manner the particle's position changes, allowing computation of \(s(10)\).

System of Equations

Solving systems of equations is key to finding the coefficients of our quadratic function. These are situations where two or more equations are solved simultaneously. In the exercise, we start with two equations derived from the conditions \(s(1) = 6\) and \(s(2) = 30\):

• \(a + b = 16\)
• \(4a + 2b = 40\)

To solve this, we use algebraic manipulation. First, both equations are aligned for ease of subtraction. By multiplying the first equation by 2, we create a pair of equations that can be subtracted to eliminate one variable. This gives us \(a = 4\), which is then substituted back to find \(b = 12\). These solutions make it possible to rewrite \(s(t) = a t^2 + b t + c\) and to determine the particle's position at various times, ensuring a comprehensive understanding of the particle's motion.