Problem 53 - Elementary Algebra

Question

The height of an object dropped from the top of a 64 -foot building is given by \(h(t)=-16 t 2+64 .\) How long will it take the object to hit the ground?

Step-by-Step Solution

Answer

The object will hit the ground in 2 seconds.

Step 1: Understand the problem

We need to determine how long it will take the object, dropped from a height of 64 feet, to hit the ground using the given equation: \( h(t) = -16t^2 + 64 \), where \( h(t) \) represents the height of the object at time \( t \) seconds.

Step 2: Set the height to zero

To find out when the object hits the ground, we set the height equation equal to zero: \( -16t^2 + 64 = 0 \). This represents the time \( t \) when the height \( h(t) \) is zero, which corresponds to hitting the ground.

Step 3: Solve the quadratic equation

Rearrange the equation to solve for \( t \): \[-16t^2 + 64 = 0 \] Add \( 16t^2 \) to both sides to get: \[ 16t^2 = 64 \]Divide both sides by 16: \[ t^2 = 4 \]

Step 4: Find the value of t

Take the square root on both sides to find \( t \): \[ t = \sqrt{4} \]This gives two possible solutions: \( t = 2 \) or \( t = -2 \). Since time cannot be negative, we choose \( t = 2 \) seconds as the valid solution.

Key Concepts

Projectile Motion

Projectile motion is a fascinating topic in physics that deals with the motion of objects thrown or propelled into the air. These motions can be split into two independent components: horizontal and vertical. For our specific exercise, the focus is primarily on the vertical motion. This is because the object is dropped directly downward.
When an object is in free fall, like in our exercise, it follows a parabolic path under the influence of gravity. Gravity is what gives the equation its distinctive form seen in the quadratic equation.

• The object starts at a certain height.
• Gravity pulls it downward, accelerating its speed.
• The height equation reflects how fast and how far it falls over time.

In the given problem, we observe that gravity's acceleration is represented by a negative factor (-16) in the equation. This negative sign indicates that the object is falling downward.

Solving Quadratic Equations

Quadratic equations are a type of polynomial equation that typically look like this:
\[ ax^2 + bx + c = 0 \]These equations often crop up in problems related to motion, like our given exercise.
To solve a quadratic equation, like the one in our exercise, here are the steps you would take:

• First, set the equation equal to zero to determine when the condition is met (like hitting the ground).
• Next, rearrange the terms logically to simplify the solving process.
• Tackle the equation by isolating the quadratic term and simplify.
• Finally, take the square root to solve for the variable.

Remember, taking the square root gives both a positive and negative answer, but in real-life applications like time, only the positive solution is viable.

Physics in Algebra

Bringing physics concepts into algebra, as we've done in this exercise, makes abstract math much more tangible. Algebra is like a language through which physics explains natural phenomena, such as motion.

By converting physical problems, like the motion of a projectile, into algebraic equations, we can systematically solve for unknowns and predict behavior. In the current exercise:

• The algebraic equation \( h(t) = -16t^2 + 64 \) describes a real-world action—dropping an object.
• The algebra lets us calculate how long it takes for the action to complete.

By interpreting and manipulating algebraic formulas, we merge mathematical theory with practical application. This dual insight is a cornerstone of disciplines requiring quantitative analysis, like engineering or physical sciences.