Problem 61 - Algebra 1

Question

Use the following information. The average speed of an object \(S\) (in feet per second) that is dropped a certain distance \(d\) (in feet) is given by the following equation. Falling object model: \(S=\frac{d}{\frac{\sqrt{d}}{4}}\) Use either equation to find the average speed of an object that is dropped from a height of 400 feet.

Step-by-Step Solution

Answer

The average speed of an object that is dropped from a height of 400 feet is 80 feet per second.

Step 1: Identify the Variables

In the given problem, the distance \(d\) that the object is dropped from is 400 feet.

Step 2: Plug in the Given Value

Substitute this value into the equation \(S=\frac{d}{\frac{\sqrt{d}}{4}}\). This gives \(S=\frac{400}{\frac{\sqrt{400}}{4}}\).

Step 3: Simplify the Equation

Now, simplify the fraction inside the main denominator. The square root of 400 is 20, so you get \(S=\frac{400}{\frac{20}{4}}\).

Step 4: Simplify Further

Divide 20 by 4 to simplify the denominator which results in \(S=\frac{400}{5}\).

Step 5: Final Calculation

Finally, divide 400 by 5 to get 80. So \(S=80\) feet per second.

Key Concepts

Average Speed

Average speed is a key concept in physics that helps us understand how fast an object moves over a distance. It is calculated by dividing the total distance traveled by the object by the total time taken to travel that distance. Average speed is different from instantaneous speed, which tells us how fast an object is moving at a specific moment in time.
In the context of our exercise, we use a formula to determine the average speed of an object that falls a certain distance. The equation provided in the problem simplifies the process by using given values, making it easier to calculate.

• Average speed gives a general idea of how quickly an object moves over a specific path.
• It is always dependent on the distance traveled and the time taken.
• For vertically falling objects, special equations like the one provided can be used since acceleration due to gravity is constant.

Understanding average speed helps us get a clearer picture of movement in a consistent and quantifiable way.

Equations

Equations are mathematical statements that assert the equality of two expressions. In mathematics and physics, they are used to represent relationships between different quantities.
The equation we used in our exercise is a special one tailored for objects falling due to gravity. The falling object model is expressed as:\[ S = \frac{d}{\frac{\sqrt{d}}{4}} \]where \( S \) is the average speed and \( d \) is the distance. This equation simplifies calculations by incorporating the square root of the distance, reflecting the influence of gravity on the falling object's speed.

• Equations help us solve practical problems by providing a mathematical framework.
• The manipulation of equations involves operations like substitution, simplification, and solving for unknowns.
• Understanding equations helps us see how different quantities are connected.

By rearranging or manipulating equations, we can isolate and solve for the desired variable, making them crucial tools in problem-solving.

Problem Solving

Problem solving in mathematics often involves identifying the problem, defining variables, and using the appropriate methods or equations to find an answer. It requires logical thinking and the ability to break a problem into smaller steps.
In our exercise, we solved the problem by following a systematic approach:

• First, identify the problem and understand what is asked; here, it was to find average speed.
• Next, identify known values and substitute them into the appropriate equation.
• Simplify the mathematical expressions step by step until reaching a final solution.

The key to problem solving is consistency and practice. By repeatedly solving different problems, one gets better at recognizing patterns and applying the most effective solutions. Each problem solved builds a foundation for tackling more complex tasks in the future.