Problem 77 - 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. A lacrosse player throws a ball upward from her playing stick from an initial height of 7 feet, with an initial velocity of 90 feet per second.

Step-by-Step Solution

Answer

The lacrosse ball takes approximately 5.6 seconds to hit the ground.

Step 1: Understanding the Vertical Motion Model

Vertical motion of an object launched upwards is represented by the equation \( h(t) = -16t^2 + v_0t + h_0 \), where \( h(t) \) is the height of the object at time \( t \), \( v_0 \) is the initial velocity, \( h_0 \) is the initial height from the ground, and -16 is the approximate acceleration due to gravity in feet per second squared.

Step 2: Application of Vertical Motion Model

In this problem, the initial height \( h_0 = 7\) feet and initial velocity \( v_0 = 90\) feet per second. The object reaches the ground when \( h(t) = 0\), so we substitute these values into the equation and set it equal to zero: \( 0 = -16t^2 + 90t + 7 \).

Step 3: Solving the Resulting Equation

We now solve the quadratic equation \(0 = -16t^2 + 90t + 7\) for \(t\). This can be done either by factoring, the quadratic formula, or by graphical methods. However, this equation won't easily factor, so we'll solve it using the quadratic formula \(t = (-b ± √(b² - 4ac)) / (2a)\), where \(a = -16, b = 90, c = 7\). Solving gives two solutions where time is greater than 0.

Step 4: Identifying the Correct Solution

We have two solutions for \(t\). However, since time can't be negative, the only solution that makes physical sense in the context of this problem is the one that is positive. That is the time taken for the lacrosse ball to hit the ground.

Key Concepts

Quadratic Equation

In the realm of mathematics and physics, a quadratic equation is an essential tool that helps us describe various types of motion, including vertical motion. This type of equation involves terms where the variable is squared. The general form of a quadratic equation is:\[a \cdot x^2 + b \cdot x + c = 0\]where \(a\), \(b\), and \(c\) are constants. In our vertical motion problem, the quadratic equation is:\[-16t^2 + 90t + 7 = 0\]attempting to solve this equation will help determine the time \(t\) it takes for an object to reach the ground. Methods to solve quadratic equations include:

• Factoring
• Graphing
• Using the quadratic formula

Though factoring can be straightforward, not all quadratic equations are easily factorable, and that's where the quadratic formula shines, providing a reliable solution for virtually any quadratic equation.

Initial Velocity

Initial velocity is the speed at which an object moves when first launched or thrown. It is a crucial component when analyzing vertical motion. In our vertical motion equation, initial velocity is represented by \(v_0\) and plays a significant role in determining how the object behaves over time.For instance, in the exercise given, the initial velocity \(v_0\) of the lacrosse ball is 90 feet per second. Initial velocity essentially represents how forceful or rapid the start of motion is.

• If initial velocity is greater, the object will reach higher altitudes.
• With a lesser initial velocity, the object might reach lower heights.

Overall, initial velocity directly influences the time it takes for the object to land, making it one of the key factors in the vertical motion model.

Initial Height

Initial height pertains to the beginning height level from which an object is launched or thrown. It is denoted by \(h_0\) in vertical motion equations and can substantially impact the time an object remains airborne.In the practice problem, the initial height \(h_0\) is 7 feet, which is the height from where the lacrosse ball is thrown. This factor will:

• Influence how long the object stays in motion before hitting the ground.
• Determine the trajectory path the object will take.

Properly acknowledging the initial height is essential, as it often is one of the starting constraints needed to solve vertical motion models. Combining it with other variables provides a complete picture of the object's journey.

Gravity

Gravity is the invisible force that pulls objects towards the Earth. In vertical motion equations, it is accounted for by the constant term and is represented as a negative acceleration owing to its downward pull.In our vertical motion formula, gravity is represented by \(-16t^2\), where 16 feet per second squared is an approximate value for gravity experienced in such scenarios.Key aspects of gravity in vertical motion models:

• It determines the rate at which the object slows down when moving upward and accelerates while coming downward.
• The negative sign indicates that gravity acts opposite to the initial upward motion.

Thus, understanding gravity's role is vital for properly analyzing and predicting the movement of objects in a vertical trajectory. Without this fundamental force, the behavior of moving objects would be vastly different.