Problem 76 - Algebra 1: Concepts and Skills

Question

Use a vertical motion model to find how long it will take for the object to reach the ground. Round your solution to the nearest tenth. An acorn falls 45 feet from the top of a tree.

Step-by-Step Solution

Answer

The time it takes for the acorn to reach the ground is approximately 1.7 seconds.

Step 1 - Identifying the Known Variables

The acorn is dropping from a height, \( h_{0} \), of 45 feet. It starts from rest, so its initial velocity, \( v_{0} \), is 0. We want to find time, \( t \), it takes to reach the ground where the height, \( h \), is 0.

Step 2 - Applying the Vertical Motion Model

The vertical motion model formula is \( h = -16t^2 + v_0t + h_0 \). Using the known variable from Step 1, replace \( h \) with 0, \( v_{0} \) with 0, and \( h_{0} \) with 45 in the model: 0 = -16t^2 + 0t + 45.

Step 3 - Solving for Time

Simplify the equation to \(16t^2 = 45 \). Then, divide both sides by 16 to get \( t^2 = 45/16 \). Then taking the square root of both sides will give us \( t = \sqrt{45/16} \).

Step 4 - Rounding to the Nearest Tenth

The time \( t \) is equal to \( \sqrt{45/16} \), which gives 1.677, roughly. Rounding this to the nearest tenth gives 1.7.

Key Concepts

Initial Velocity

Initial velocity, often represented as \( v_0 \), is a crucial concept in understanding vertical motion. In this context, initial velocity refers to the speed of an object when it begins its journey. In many physics problems, especially those involving vertical motion, initial velocity is the starting speed before any forces – like gravity – further influence the object.
For instance:

• If an object is dropped from a height, like an acorn falling from a tree, and it starts from rest, its initial velocity is 0.
• If the object was thrown, its initial velocity would be the speed at which it was launched.

Understanding initial velocity helps in predicting how an object will move. It sets the scene for further calculations, as initial velocity directly impacts the object's motion as described by the vertical motion model equation.

Quadratic Equations

Quadratic equations appear frequently when dealing with vertical motion problems. These equations are characterized by their highest term being squared, commonly looking like \( ax^2 + bx + c = 0 \).
In vertical motion:

• The equation takes the form \( h = -16t^2 + v_0t + h_0 \). Here, \(h\) is the height, \(-16t^2\) arises from gravity's influence, \(v_0t\) is the initial velocity term, and \(h_0\) is the starting height.
• Solving these equations typically involves getting them into the standard form, then applying algebraic methods to find the value of the variable \( t \) (time, in this context).

Quadratic equations help predict when an object reaches a certain height; in this example, when an acorn hits the ground. Thus, they are essential for solving vertical motion problems.

Solving Equations

Solving equations is the mathematical process employed to find the variable values in a given equation. When addressing vertical motion problems like our acorn example, you're often tasked with finding how long it takes an object to reach the ground.
Steps to solve such equations include:

• Start by substituting known values into the vertical motion model. Here, \(h = 0\), \(v_0 = 0\), and \(h_0 = 45\).
• This results in a simplified version of the quadratic equation: \( -16t^2 + 45 = 0 \).
• Through algebraic manipulation, isolate the \( t^2 \) term, then solve for \( t \) by taking the square root.

Finally, rounding is often necessary to provide a practical answer. For example, in our case, \( t = \sqrt{\frac{45}{16}} \) approximately equals \( 1.7 \) seconds, so the acorn reaches the ground in this time. Understanding these steps is critical to effectively tackle similar physics problems.