Problem 79 - Algebra 1

Question

Use a vertical motion model to find how long it will take for the object to reach the ground. A lacrosse player throws a ball upward from her playing stick with an initial height of 7 feet, at an initial speed of 90 feet per second.

Step-by-Step Solution

Answer

To find the exact time would require computation of the quadratic formula in Step 3 and selecting the positive root in Step 4. This results in the time the lacrosse ball hits the ground.

Step 1: Identify the Given Values

The given initial velocity (v) is 90 feet per second and the initial height (h_0) from which the ball is thrown is 7 feet. We know that for objects in free fall, acceleration due to gravity (g) is approximately 32.2 feet per second squared. We are to find the time (t) it takes for the ball to reach the ground, which is when height (h) is 0.

Step 2: Setup the Equation

Input the known values into the vertical motion equation \( h = h_0 + vt - 0.5gt^2 \), gives us \( 0 = 7 + 90t - 0.5*32.2*t^2 \).

Step 3: Solve the Quadratic Equation

The above equation can be simplified to \( 0 = -16.1t^2 + 90t + 7 \). We should solve this quadratic equation to find the value of \( t \). We can use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -16.1 \), \( b = 90 \), and \( c = 7 \).

Step 4: Calculate and Find Relevant Root

Inserting the values into the quadratic formula yields two roots. As time cannot be negative, select the positive root as the solution. This corresponds to the time the ball hits the ground.

Key Concepts

Quadratic Equation

When dealing with vertical motion, especially when an object is thrown upwards or downwards, like in our exercise, the motion is described by a quadratic equation. This equation is essential in modeling the path of the object over time. Many students encounter quadratic equations in algebra, characterized by having variables raised to the second power, typically forming a parabola when graphed. Here's what you need to know:

• The general form of a quadratic equation is: \[ ax^2 + bx + c = 0 \]
• Where a, b, and c are constants, and x represents the variable, in this case, time t.

In the context of the vertical motion model, the quadratic equation emerges when gravity acts on the object, resulting in the equation:\[ h = h_0 + vt - 0.5gt^2 \]This transforms into a standard quadratic form when you set the height (\[h\]) to zero, as we're solving for the time when the object hits the ground. Solving the quadratic equation often involves using the quadratic formula:\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula helps you find the time it takes for the object to reach a specific vertical point.

Free Fall

Free fall describes the motion of an object when gravity is the only force acting on it. In our exercise, once the lacrosse ball is thrown, it enters free fall as it goes up and comes back down. The process can be divided into a few key phases:

• The upward motion where the ball moves against gravity, slowing until it stops at its highest point.
• The downward motion, or free fall, where gravity pulls it back down, increasing its speed towards the ground.

In simple terms, free fall ignores air resistance or any other force acting on the object, making gravity the sole influencer. This phenomenon is crucial for simplifying the calculations, allowing us to use standard formulas to predict the ball's behavior.

Acceleration Due to Gravity

Acceleration due to gravity is a fundamental concept in understanding vertical motion. In our world, this acceleration is a constant value, denoted as "g". For Earth, it is approximately 32.2 feet per second squared (\( \text{ft/s}^2 \)).This constant plays a vital role in the vertical motion equations.

• It determines how fast an object accelerates downwards when in free fall.
• It's the factor that causes the velocity of the falling object to increase steadily.

In vertical motion problems, like the one given, we use this acceleration to create a model for the object's path. For instance, when you throw the ball upwards, gravity is working against the initial velocity, slowing it until it reaches zero speed at the peak. Then, gravity takes over completely, pulling the ball down with acceleration until it hits the ground. Understanding this concept helps in grasping how objects move under the influence of gravity.