Problem 48 - Precalculus with Limits

Question

Vertical Motion In Exercises 47 and \(48 ,\) an object moving vertically is at the given heights at the specified times. Find the position equation \(s = \frac { 1 } { 2 } a t ^ { 2 } + v _ { 0 } t + s _ { 0 }\) for the object. $$ \begin{array} { l } { \text { At } t = 1 \text { second } , s = 132 \text { feet } } \\ { \text { At } t = 2 \text { seconds, } s = 100 \text { feet } } \\\ { \text { At } t = 3 \text { seconds, } s = 36 \text { feet } } \end{array} $$

Step-by-Step Solution

Answer

After completing the steps above, the position equation of the object will be achieved according to the values of \(a\), \(v_0\), and \(s_0\) found.

Step 1: Formulate Equations

For \(t = 1\) second, \(s = 132\) feet. Substituting into the equation \(s = \frac{1}{2}at^2 + v_0t + s_0\), we get \(132 = \frac{1}{2}a + v_0 + s_0\) (equation 1). For \(t = 2\) seconds, \(s = 100\) feet. Substituting into the equation we get \(100 = 2a + 2v_0 + s_0\) (equation 2). For \(t = 3\) seconds, \(s = 36\) feet. Substituting into the equation we get \(36 = \frac{9}{2}a + 3v_0 + s_0\) (equation 3).

Step 2: Solve the Equations

We have a system of three equations that we now need to solve to find the values of acceleration \(a\), initial speed \(v_0\), and initial position \(s_0\). Substract equation 1 from equation 2 to find an express for \(a\), then substitute \(a\) into equation 1 to find \(v_0\), and finally substitute both \(a\) and \(v_0\) into equation 1 to find \(s_0\).

Step 3: Formulate the Position Equation

Once we have the values for acceleration, initial speed, and initial position, we substitute these back into the position equation \(s = \frac{1}{2}at^2 + v_0t + s_0\).

Key Concepts

Position Equation

The position equation is an essential tool in answering questions about vertical motion. It helps you determine the height or position of an object at any given time when you know the acceleration, initial speed, and initial position. The general formula for vertical motion is given by:

• \[s = \frac{1}{2}at^2 + v_0 t + s_0\]

Here, \(s\) represents the position or height of the object at time \(t\). This equation is particularly useful for objects thrown straight up or moving under the influence of gravity.
When tackling problems involving vertical motion, substituting given values of time and position into this equation can help solve for unknowns like acceleration, initial speed, or initial position. Understanding how to manipulate this equation is crucial for finding solutions in various motion-related problems.

Acceleration

Acceleration is a measure of how quickly an object's velocity changes over time. In vertical motion, acceleration is often due to gravity, which pulls objects towards the Earth's surface at a constant rate.

• In many problems, especially those involving free fall or objects projected vertically, acceleration is typically given as \(-9.8 \text{ m/s}^2\) when working in metric units or \(-32 \text{ ft/s}^2\) when working in imperial units.
• In the position equation, acceleration is represented by the letter \(a\). It plays a critical role in determining how an object's speed and position change over time.

When solving for acceleration using the position equation, compare multiple time-position instances. By forming equations for different times and solving them, you can isolate \(a\) and find its value. This approach is particularly handy when acceleration is not explicitly given, requiring you to deduce it from other information.

Initial Speed

Initial speed, denoted as \(v_0\), is the speed at which an object starts its motion before any force of acceleration (like gravity) acts on it. It sets the stage for how fast the object will move vertically.

• A higher initial speed means the object will travel upwards longer before gravity slows it down and eventually pulls it back down.
• In problems involving vertical motion, determining \(v_0\) is key to understanding how the object's velocity changes over time.

While this speed is often given directly, there are cases where you have to calculate it using the position equation and known positions at various times. By solving the system of equations obtained from substituting known values of \(t\) and \(s\) into the position equation, you can figure out the initial speed if it's unknown.

Initial Position

The initial position, represented by \(s_0\), is the height or position of an object at \(t = 0\). It serves as the starting point in any vertical motion problem.

• It tells you where the object begins its journey, and it's crucial when applying the position equation.
• Whether the object is launched from the ground, a certain height, or even a moving platform, knowing \(s_0\) helps contextualize the object's motion.

When solving the position equation, initial position is an essential factor. Even if \(s_0\) is not initially given, it can be solved by using the system of equations derived from various positions at different times. Ensuring you accurately calculate or identify the correct initial position is vital for accurate results in any analysis involving vertical motion.