Problem 79 - Algebra 1

Question

A boulder falls off the top of a cliff during a storm. The cliff is 60 feet high. Find how long it will take for the boulder to hit the road below. Write the falling object model for \(s=60\)

Step-by-Step Solution

Answer

The time it will take for the boulder to hit the ground is \(\sqrt{60 / 16}\) seconds.

Step 1: Identify the Given and the Unknown

The given: \(s = 60\) feet, \(g = 32\) feet/second². The unknown: \(t\) (the time to hit the ground).

Step 2: Apply the Falling Object Model

The formula for the falling object is \(s = 1/2gt^2\). Substituting the given values, we get \(60 = 1/2 * 32 * t^2\).

Step 3: Rearrange the Formula to Solve for \(t\)

To solve for \(t\), divide both sides of the equation by \(16\) (1/2 of 32). We get \(t^2 = 60 / 16\). Therefore, \(t = \sqrt{60 / 16}\).

Step 4: Calculate the Value of \(t\)

Calculate the right side to find \(t\).

Key Concepts

Kinematics in One Dimension

Understanding kinematics in one dimension is crucial for analyzing the motion of objects moving in a straight line, such as a boulder falling off a cliff. In this scenario, the key variables include the initial position, the final position (height of the cliff), and the acceleration due to gravity. With these variables, kinematics allows us to describe the motion of the falling boulder without concern for the mass of the object or factors like air resistance.

For our exercise, using the falling object model is appropriate as it simplifies the situation to one dimension—vertical motion under the influence of gravity. To solve for the time it takes for an object to hit the ground, we use the equation for displacement in the form of \( s = \frac{1}{2}gt^2 \), where \( s \) is the distance fallen, \( g \) is the acceleration due to gravity, and \( t \) is the time taken to fall that distance.

Quadratic Equations

When solving problems involving objects in free fall, we often encounter quadratic equations. A typical equation might look something like \( s = \frac{1}{2}gt^2 \), which can be rearranged into the standard quadratic form \( at^2 + bt + c = 0 \).

Quadratic equations are fundamental in algebra and can be solved using various methods, such as factoring, completing the square, using the quadratic formula, or by extracting square roots when applicable. In our exercise, by setting \( s = 60 \) feet and assuming no initial velocity, the formula becomes a simple quadratic equation \( 60 = \frac{1}{2} * 32 * t^2 \) that we can solve for \( t \) by isolating the variable and taking the square root after simplifying the equation.

Solving for Time

Solving for time in physics involves rearranging equations to express time as a function of other known variables. In the context of the falling object model, we're dealing with the square of time \( t^2 \), which requires that we take the square root after isolating \( t \) on one side. When rearranging the formula, it’s important to follow algebraic conventions carefully, which means balancing equations by performing operations on both sides.

In the falling boulder problem, by dividing both sides by \( 16 \) (\( \frac{1}{2} \) of \( 32 \) feet/second²), we get \( t^2 = \frac{60}{16} \) and then \( t = \sqrt{\frac{60}{16}} \). Here, solving for time is the final step in understanding when the boulder will impact the road, and it involves correctly applying arithmetic operations to compute the value.

Physics in Algebra

Physics often employs algebra to create models and solve real-world problems. By using algebraic symbols to represent physical quantities, we can create equations that describe physical phenomena, such as the motion of falling objects. The beauty is in converting conceptual understanding (like the effect of gravity on an object in free fall) into mathematical expressions that can be manipulated to find specific values, such as time and distance.

For the falling boulder, algebra helps us manipulate the falling object model equation to solve for the unknown time. This intersection of physics and algebra demonstrates how mathematical tools are applied within the physical sciences to predict outcomes and analyze events. In our case, the algebraic manipulation of the equation allows us to predict precisely how long it takes for the boulder to reach the ground.