Problem 79 - Algebra and Trigonometry Real Mathematics, Real People

Question

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. At \(t=1\) second, \(s=352\) feet. At \(t=2\) seconds, \(s=272\) feet. At \(t=3\) seconds, \(s=160\) feet.

Step-by-Step Solution

Answer

The position equation for the object is \(s=16t^{2}-96t+416\).

Step 1: Set Up the Equations

We have three times with corresponding positions meaning we can set up three equations for each. Using \(s=\frac{1}{2} a t^{2}+v_{0} t+s_{0}\), the equations are: At \(t=1\) second, \(352=\frac{1}{2} a +v_{0} +s_{0}\), At \(t=2\) seconds, \(272=a +2v_{0} +s_{0}\) and at \(t=3\) seconds, \(160=\frac{3}{2} a +3v_{0} +s_{0}\).

Step 2: Solve the system of equations

Subtract the first equation from the second and third equations. This will give two new equations: \(272-352=a +2v_{0} +s_{0}-(\frac{1}{2} a +v_{0} +s_{0})\), which simplifies to: \(-80=\frac{1}{2}a +v_{0}\) and \(160-352=\frac{1}{2} a +3v_{0} +s_{0}-(\frac{1}{2} a +v_{0} +s_{0})\), which simplifies to: \(-192=2v_{0}\). This gives \(v_{0}=-96\) and substituting \(v_{0}=-96\) into \(-80=\frac{1}{2}a +v_{0}\) will give \(a=32\). We then substitute \(a=32\) and \(v_{0}=-96\) into the first equation, \(352=\frac{1}{2} a +v_{0} +s_{0}\) to get \(s_{0}=352-\frac{1}{2} *32 -(-96)=416\).

Step 3: Fill in the constants

Now we can go back to the original equation and fill in the constants we found: \(a=32\), \(v_{0}=-96\), \(s_{0}=416\). This gives the final equation of motion as \(s=\frac{1}{2} * 32t^{2}-96t+416\).

Key Concepts

Equation of Motion

Understanding the equation of motion is essential when analyzing the vertical motion of objects. The basic form is given by:

• \( s = \frac{1}{2} a t^2 + v_0 t + s_0 \)

In this equation:

• \(s\) represents the position of the object at time \(t\).
• \(a\) is the acceleration, a key factor, especially in free-fall problems where it can be due to gravity.
• \(v_0\) is the initial velocity, indicating how fast the object was moving at the start.
• \(s_0\) is the initial position, or where the object started before it began moving.

To use the equation effectively, you must input the correct values for \(a\), \(v_0\), and \(s_0\). Insights into the object's motion can be calculated once these constants are determined. These can be found by setting up and solving multiple position-time equations with various points in time, as was shown with different time instances in the original exercise.

System of Equations

A system of equations arises when we have multiple interrelated equations that share the same variables. In our exercise, because the object's position at different times was given, we formed a system of three equations to find unknowns.

The Importance

Solving such systems allows us to discover unknown variables by using the relationships between equations. In the vertical motion scenario, solving the system gave us insights about acceleration and velocity.

How to Solve

• First, express each given scenario as an equation based on the equation of motion.
• Next, manipulate these equations to eliminate variables and find solutions.
• Start by subtracting one equation from another to simplify and solve for one variable.

In this case, values for initial velocity \(v_0\), acceleration \(a\), and starting position \(s_0\) were found sequentially.

Vertical Motion

Vertical motion involves objects moving up or down, typically under the influence of gravity. In physics, analyzing this type of motion assumes air resistance is negligible and only gravity acts on the object.

Key Aspects of Vertical Motion

• Gravity's Influence: It's a force constantly acting downward, and in most scenarios, the acceleration due to gravity is approximated as \(9.8 \, m/s^2\) or \(32 \, ft/s^2\).
• Free-Fall Scenario: Either when an object is dropped or thrown in the air without resistance.
• Equation Application: Accurately using the equation of motion helps in predicting the object's future position at any time \(t\).

Using these concepts, you can predict how an object, such as a ball or a rocket, behaves under the sky. Values determined from solving our system of equations were then applied to predict the exact path and positioning of a vertically moving object.