Problem 65 - Algebra A Combined Function

Question

The formula \(v=\sqrt{2 g h}\) relates the velocity \(v\), in feet per second, of an object after it falls \(h\) feet accelerated by gravity \(g\), in feet per second squared. If \(g\) is approximately 32 feet per second squared, find how far an object has fallen if its velocity is 80 feet per second.

Step-by-Step Solution

Answer

The object has fallen 100 feet.

Step 1: Identify Known Quantities

We know the velocity, \(v = 80\) feet per second, and the gravitational acceleration, \(g = 32\) feet per second squared. We need to find the distance \(h\) that the object has fallen.

Step 2: Rearrange the Formula to Solve for h

The original formula is \(v = \sqrt{2gh}\). To find \(h\), first square both sides to eliminate the square root, giving us \(v^2 = 2gh\). Then, solve for \(h\) by dividing both sides by \(2g\), resulting in \(h = \frac{v^2}{2g}\).

Step 3: Substitute Values into the Rearranged Formula

Substitute \(v = 80\) and \(g = 32\) into the formula: \[h = \frac{80^2}{2 \times 32}\].

Step 4: Calculate the Distance h

First, calculate \(80^2 = 6400\). Then, calculate the denominator \(2 \times 32 = 64\). Finally, divide the numerator by the denominator: \(h = \frac{6400}{64}\).

Step 5: Final Answer for h

Perform the division: \(h = 100\). Thus, the object has fallen 100 feet.

Key Concepts

Kinematics

Kinematics is a branch of physics that describes the motion of objects without considering the forces that cause this motion. It involves understanding concepts such as displacement, velocity, acceleration, and time. In simple terms, kinematics allows us to predict how an object will move based on initial conditions and mathematical relationships.

Key concepts in kinematics include:

• Displacement: the change in position of an object.
• Velocity: the rate of change of displacement with time.
• Acceleration: the rate of change of velocity with time.
• Equations of motion: these are equations that relate the above quantities.

Kinematics is crucial for analyzing problems like the one in the original exercise, where we're interested in finding out how far an object falls based solely on its velocity.

Free Fall

Free fall describes the motion of an object under the influence of gravitational force only, without any air resistance. In the context of physics, when an object is in free fall, the only force acting upon it is gravity. This causes the object to accelerate, increasing its velocity as it continues to fall.

Key characteristics of free fall include:

• The initial velocity might be zero if the object starts dropping from rest.
• The object experiences a constant acceleration due to gravity.
• The equations of motion can simplify because the only force is gravity.

Understanding free fall helps in deriving relationships like the one given in the original formula, where velocity and height are connected through gravitational acceleration.

Gravitational Acceleration

Gravitational acceleration is the acceleration imparted to objects due to earth's gravity. It is approximately 9.8 meters per second squared (or about 32 feet per second squared in the imperial system used here). This acceleration is constant and acts toward the center of the Earth.

Important facts about gravitational acceleration:

• It is denoted as "g" in equations.
• It affects all objects equally, regardless of their mass.
• In free fall, gravitational acceleration is the only acting force.

In the original exercise, understanding gravitational acceleration is key to solving for the distance the object falls since it directly influences the velocity gained during fall.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to isolate the variable of interest. It's a skill frequently used in physics to make predictions or solve for unknowns.

Techniques used in algebraic manipulation:

• Squaring both sides of an equation to eliminate square roots.
• Dividing or multiplying both sides by the same factor to simplify equations.
• Substituting known values into equations to find unknowns.

In the provided solution, algebraic manipulation is the core process that allowed the conversion of the initial velocity formula into one that solves for height, giving us the desired result after calculations.