Problem 53 - Introductory Algebra for College Students

Question

A football is kicked straight up from a height of 4 feet with an initial speed of 60 feet per second. The formula,$$h=-16 t^{2}+60 t+4$$ describes the ball's height above the ground, \(h\), in feet, \(t\) seconds after it is kicked. How long will it take for the football to hit the ground? Use a calculator and round to the nearest tenth of a second.

Step-by-Step Solution

Answer

Thus, the ball will hit the ground approximately 3.8 seconds after it is kicked.

Step 1: Identify the coefficients for the quadratic equation

The given equation \(h=-16 t^{2}+60 t+4\) is in the form of a quadratic equation \(ax^{2} + bx + c = 0\). Here \(a = -16\), \(b = 60\), and \(c = 4\).

Step 2: Set the equation to 0 and solve for \(t\)

Evoking the question, we know the ball has hit the ground when \(h = 0\). Hence, the equation becomes 0 = -16 \(t^{2}\) + 60\(t\) + 4.

Step 3: Solve the equation using the quadratic formula

The quadratic formula is \(-b± √(b^2-4ac) / 2a\). Plugging in the values from the above equation, we get \(t = [-60± √(60^2-4*(-16)*4)] / 2*(-16)\).

Step 4: Simplify to get 't' values

This yields two values for \(t\), one when considering plus sign and one with the minus sign in \(-60± √(3600+256) / -32\). Positive time is physically possible, hence take the value with the 'plus'. Simplified, we get \(t ≈ -(-60-62) / -32 = 3.8\) seconds.

Key Concepts

Projectile Motion

Projectile motion refers to the movement of an object thrown or projected into the air. It is subject to acceleration due to gravity. This type of motion occurs in two dimensions, usually horizontal and vertical. In the case of our exercise, the football represents the projectile.The typical characteristics of projectile motion include:

• Initial velocity: This is how fast the object begins its motion. For the football, it was kicked up with a speed of 60 feet per second.
• Time of flight: This is how long the projectile is in the air. We are asked to calculate the time it takes for the football to hit the ground.
• Maximum height: The highest point the projectile reaches. Even though we do not find this in the problem, it's a key part of all projectile motions.
• Range: This is how far the projectile travels. In the context here, it refers to distance upwards, influenced by initial height and speed.

Solving problems involving projectile motion often requires understanding how to manipulate the formulas that describe vertical displacement, like the one in the exercise: \(h=-16t^{2}+60t+4\). The key is recognizing the effects of gravity, which is represented by the term \(-16t^{2}\). This formula is crucial for finding out when the projectile (the football in this case) returns to the starting height, or in our problem, when it hits the ground.

Solving Quadratic Equations

A quadratic equation is any equation that can be rearranged in standard form as \(ax^{2} + bx + c = 0\). In this exercise, the equation \(-16t^{2} + 60t + 4 = 0\) serves as our quadratic equation.To solve quadratic equations, several methods can be applied:

• Factoring: This involves rewriting the equation as the product of simpler expressions. Not always possible for equations with decimal solutions.
• Completing the square: It makes the expression a perfect square trinomial. Useful for deriving the quadratic formula or in teaching logic about quadratics.
• Quadratic formula: A versatile tool that works for any quadratic equation. This method is highlighted in our exercise.

We use the quadratic formula here because of its reliability and efficiency in yielding solutions. Set the expression to zero when you want to find specific points where the calculations align with real-world outcomes, like a projectile hitting the ground.

Quadratic Formula

The quadratic formula is a vital mathematical tool used to find the roots of a quadratic equation. This formula is given by:\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]In the context of the exercise involving a football's trajectory, the formula helps determine the time at which the ball will hit the ground. Let's break it down into easier steps:

• -b: The negative of the coefficient of the linear term. In our equation, this is -60.
• \(b^2 - 4ac:\) Known as the discriminant. It determines the number and type of roots. Here, it is \(60^2 - 4(-16)(4)\), simplifying to \(3600 + 256\).
• 2a: Twice the coefficient of the quadratic term. For us, 2(-16) gives -32.

Using these values, the formula provides two solutions. However, only positive physical values of \(t\) make sense for time, thus leading us to select the appropriate value of \(t = 3.8\) seconds for when the ball hits the ground. It shows how mathematical principles apply practically, allowing us to predict motion and other real-world phenomena effectively.