Problem 22 - Algebra 2 and Trigonometry

Question

The height \(h,\) in feet, of a golf ball shot upward from a ground level sprint gun is described by the formula \(h=-16 t^{2}+48 t\) where \(t\) is the time in seconds. When will the ball hit the ground again?

Step-by-Step Solution

Answer

The ball will hit the ground again at \(t = 3\) seconds.

Step 1: Understand the Problem

We need to find the time \(t\) when the golf ball will hit the ground again. This means we want to find when the height \(h\) is equal to 0.

Step 2: Set up the Equation

Since the golf ball hits the ground when \(h = 0\), use the formula: \[-16t^2 + 48t = 0\]

Step 3: Factorize the Equation

Factor the equation by finding the common factor: \[-16t(t - 3) = 0\]

Step 4: Solve for t

Set each factor to zero: 1. \(-16t = 0\) 2. \(t - 3 = 0\)

Step 5: Find the Solution for t from Each Equation

1. From \(-16t = 0\), we solve for \(t\):\[t = 0\]2. From \(t - 3 = 0\), we solve for \(t\):\[t = 3\]

Step 6: Interpret the Solutions

The solutions are \(t = 0\) and \(t = 3\). \(t = 0\) represents the initial time when the ball is launched. Thus, \(t = 3\) seconds is when the ball will hit the ground again.

Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object that is projected into the air and is influenced only by the force of gravity and its initial velocity. The path it follows is typically a parabola. In the example of the golf ball shot from the ground, we see projectile motion in action. The golf ball goes up into the air and then, under the influence of gravity, comes back down to the ground.

The height at any time during its flight can be described using quadratic equations. For the golf ball, its height over time is given by the formula: \(h = -16t^2 + 48t\).

Here's what happens during projectile motion:

• The ball is launched with an initial velocity that gives it upwards motion, captured by the first part of the formula \(48t\).
• Gravity, a constant force pulling objects towards the earth, is represented by the \(-16t^2\) term, which reduces the height over time.

Understanding projectile motion is essential when solving problems related to objects in flight, as it helps you predict where and when they will land.

Factoring Quadratics

Factoring quadratics is a key step in solving quadratic equations, which often appear in projectile motion problems. When we factor a quadratic equation, we express it as a product of its simpler, lower-degree polynomials. This process often simplifies the process of finding the equation's solutions.

For the equation \(-16t^2 + 48t\), we can start by looking for common factors. Notice that \(-16t\) is a factor common to both terms in the equation. Extracting this, we simplify the equation to: \[-16t(t - 3) = 0\]

Factoring quadratics helps in breaking down the problem into manageable steps:

• Identify a common factor from all terms.
• Use the distributive property to rewrite the equation.
• Set each factor equal to zero to find solutions.

By recognizing patterns in quadratic equations, factoring becomes more intuitive, allowing for quicker solutions.

Solving for Variables

Once a quadratic is factored, the next step is solving the equation to find the variables that satisfy it. This involves setting each factored term equal to zero, because anything multiplied by zero results in zero. For our example equation \(-16t(t - 3) = 0\), we split it into two smaller equations, which are easier to solve:

• \(-16t = 0\)
• \(t - 3 = 0\)

Solving these gives:

• For \(-16t = 0\), divide both sides by \(-16\) to find \(t = 0\).
• For \(t - 3 = 0\), add 3 to both sides to find \(t = 3\).

These solutions tell us the times when certain conditions are true. Here, \(t = 0\) indicates the time the ball was launched, and \(t = 3\) indicates when it hits the ground again. Solving for variables is crucial in determining outcomes in equations involving projectile motion or any scenario requiring quadratic equations.