Problem 59 - Algebra 1: Concepts and Skills

Question

A boulder falls off the top of an overhanging cliff during a storm. The cliff is 96 feet high. Find how long it will take for the boulder to hit the road below. Write a falling object model when \(s=96\)

Step-by-Step Solution

Answer

It will take approximately \( \sqrt{6} \approx 2.45 \) seconds for the boulder to hit the road below.

Step 1: Understand the problem

The overhanging cliff is 96 feet high and a boulder falls off the top to hit the road. We use the rule of falling objects where the distance an object has fallen is given by the formula \(s = 16t^2\) where s is the distance in feet, t is the time in seconds.

Step 2: Input the given distance

The distance given is \(s = 96\) feet, so substitute \(s = 96\) in the formula to get \(96 = 16t^2\).

Step 3: Solve for t

Divide both sides of the equation by 16 to solve for \(t^2\). We get \(t^2 = \frac{96}{16} = 6\). Then take square root of \(6\) to find \(t\). We get \(t = \sqrt{6}\).

Key Concepts

Quadratic Equations

In this problem, we use a quadratic equation to determine the time it takes for a boulder to fall from a cliff to the ground. A quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \). In our context, the equation is derived from the falling object model given by \( s = 16t^2 \). Here, \( s \) is the distance the object falls, and \( t \) is time in seconds. This equation can be rearranged into the standard quadratic form by setting \( s - 16t^2 = 0 \), where 16 is the coefficient of \( t^2 \). To solve such equations and find \( t \), you often need to isolate \( t^2 \) and then find \( t \) by taking the square root. Quadratic equations like these arise frequently in physics when analyzing motion under constant acceleration, such as gravity.

Physics Applications in Math

The falling object model elegantly combines principles from physics and mathematics. It shows how mathematical tools can model physical phenomena accurately. The equation \( s = 16t^2 \) is a simplified expression derived from the kinematic equations for objects in free fall under gravity, specifically for motion near the Earth's surface. Physics often involves calculating how an object moves over time, how fast it goes, and how far it travels. By using mathematical equations, we can precisely compute these values.

• Mathematics provides the structure and language to describe physical laws.
• Physics gives meaning and context to the mathematical calculations.

The close relationship between math and physics helps us to predict and understand real-world behaviors, such as a boulder falling.

Gravity in Motion

Gravity is the force acting on an object in free fall, pulling it towards Earth. In the falling object model \( s = 16t^2 \), the coefficient 16 reflects gravitational acceleration in feet per second squared (approx 32 feet per second squared for Earth's gravity, divided by 2). This model assumes no air resistance, providing a clean example of constant acceleration. Understanding how gravity impacts objects in motion helps explain a variety of phenomena. For example:

• Objects accelerate as they fall, increasing speed over time.
• The time taken to reach the ground is dependent on the height from which they fall.
• Mathematical models of gravity are crucial in fields like engineering and aerospace.

By analyzing motion with gravity, we gain insight into how forces affect movement, providing valuable application in real-world scenarios.