Problem 99 - Elementary Algebra

Question

The height in feet of an object dropped from a 9-foot ladder is given by \(h(t)=-16 t 2+9,\) where \(t\) represents the time in seconds after the object has been dropped. How long does it take the object to hit the ground? (Hint: The height is 0 when the object hits the ground.)

Step-by-Step Solution

Answer

It takes \( \frac{3}{4} \) seconds for the object to hit the ground.

Step 1: Understanding the Problem

We are given the height equation of a falling object, \( h(t) = -16t^2 + 9 \). We need to find the time \( t \) when the object hits the ground. According to the hint, the object hits the ground when \( h(t) = 0 \).

Step 2: Set Up the Equation

Replace \( h(t) \) with 0 in the given equation to set up the equation: \( 0 = -16t^2 + 9 \).

Step 3: Move Terms

Reorganize the equation by moving terms to set one side to zero: \( 16t^2 = 9 \).

Step 4: Divide Both Sides

Divide both sides of the equation by 16 to isolate \( t^2 \): \( t^2 = \frac{9}{16} \).

Step 5: Take the Square Root

Take the square root of both sides to solve for \( t \): \( t = \sqrt{\frac{9}{16}} \).

Step 6: Simplify the Square Root

Simplify \( \sqrt{\frac{9}{16}} \) to \( t = \frac{3}{4} \), since the square root of 9 is 3 and the square root of 16 is 4.

Key Concepts

Free fall motion

Free fall motion is a fascinating aspect of physics that deals with the movement of objects solely under the influence of gravity. When an object is in free fall, it means that gravity is the only force acting upon it, thus ignoring air resistance for simplicity. In the context of our problem, a 9-foot ladder object falls without anything hindering its motion, other than gravity pulling it down.

Free fall specifically follows these general rules:

• The object starts from rest, meaning its initial velocity is zero.
• It experiences constant acceleration due to gravity, which on Earth is approximately \( g = 32 \, \text{ft/s}^2 \).
• The position (height in this scenario) of the object as a function of time can be modeled with quadratic equations.

Each of these principles of free fall helps us predict how long it will take for an object to hit the ground, hence the importance of comprehending the height equation.

Solving quadratic equations

Quadratic equations are a kind of polynomial equation of the form \( ax^2 + bx + c = 0 \). They are fundamental in algebra and are prominent in modeling several real-world scenarios, including free fall.

In this exercise, we have a quadratic equation derived from the height formula, \( h(t) = -16t^2 + 9 \). The exercise involves solving this equation to find out when the object will hit the ground, which means setting the equation to zero \( 0 = -16t^2 + 9 \).

To solve quadratic equations, we can use various methods such as:

• Factoring
• Completing the square
• Quadratic formula
• Square root method (used in this problem)

The quadratic nature of the equation means the solution represents the time it takes for the object to hit zero height, or the ground in this case.

Square root method

The square root method is one of the straightforward techniques for solving quadratic equations, especially when the equation is in the form \( ax^2 = c \). It requires isolating the square term and then taking the square root of both sides.

For example, in the solution:

• We start by reorganizing the equation \( 16t^2 = 9 \).
• We then divided both sides by 16, giving \( t^2 = \frac{9}{16} \)
• Taking the square root of both sides yields \( t = \sqrt{\frac{9}{16}} \).
• Simplifying gives \( t = \frac{3}{4} \).

In this context, \( t \) represents time, and by using the square root method, we find the object takes \( \frac{3}{4} \) seconds to reach the ground. This method is efficient and effective when the quadratic is perfect for simplifying after isolating the square term.