Problem 16 - Algebra 1: Concepts and Skills

Question

Use the falling object model, \(h=-16 t^{2}+s .\) Given the initial height \(s\), find the time it would take for the object to reach the ground, disregarding air resistance. Round the result to the nearest tenth. \(s=160\) feet

Step-by-Step Solution

Answer

The time it would take for the object to reach the ground is approximately 3.2 seconds.

Step 1: Substitute the values into the equation

First, substitute the given values into the falling object model equation. Here, \(s=160\) feet and \(h=0\) (as the object has reached the ground), so the equation becomes: \(0=-16 t^{2}+160 \).

Step 2: Solve for \(t^{2}\)

Next, rearrange the equation to solve for \(t^{2}\). The equation therefore becomes: \(16 t^{2}=160\).

Step 3: Solve for t

To solve for t, take the square root of both sides of the equation. But first, simpler the right hand side by dividing by 16 which yields \(t^{2}=10\). Then, \(t=\sqrt{10}\), which is approximately 3.2. Remember, negative root doesn't make sense in this context since time must be non-negative.

Key Concepts

Understanding Quadratic Equations

Quadratic equations take the form of \( ax^2 + bx + c = 0 \). These equations might look complex at first, but they're very manageable once you break them down.
In the context of our falling object problem, the equation used is a specific type of quadratic equation. Here, the standard form is simplified to \(-16t^2 + s = 0 \), where:

• \(a = -16\)
• \(b = 0\)
• \(c = s\) which is the initial height.

Instead of all the letters, we focus on time \(t\) and the initial height \(s\).
In simpler terms, the quadratic equation helps us understand how time and height relate for a falling object, under the influence of gravity.
If you have values plugged in (like the initial height \(s\)), you can use algebraic manipulations to find time \(t\), showing when the height \(h\) reaches zero, or the ground.

Understanding Initial Height

The initial height refers to the starting vertical position of an object before it begins its descent. In our falling object model, this is denoted by \(s\).
Step by step, think of the initial height as the point from which the object is dropped or released.
This height is crucial for calculations because it is used to determine when the object will reach the ground.
For our exercise involving a falling object from \(160\) feet, the initial height \(s = 160\) becomes the key value in the quadratic equation.
It's important because it sets the stage for the rest of the calculations. Since we want to find out when the object hits ground level (where \(h = 0\)), knowing the initial height directly affects the time of flight calculated.

Learning About Time of Flight

Time of flight is the duration it takes for a falling object to travel from its initial height down to a specific point, in this scenario, the ground level.
The time of flight calculation typically involves factoring in initial height and the constant acceleration due to gravity.
In our problem, "time of flight" is extracted from the quadratic equation \(-16 t^2 + s = 0\). Solving this equation allows us to figure out how long it takes for the object to fall from \(160\) feet to \(0\) feet.
Once simplified, the equation allows for \(t^2 = 10\), leading to \(t = \sqrt{10} \), approximately \(3.2\) seconds.
Remember, time of flight does not consider upward force, like air resistance, in this model, making calculations straightforward yet quite idealized.