Problem 19 - College Algebra Essentials

Question

A football was kicked vertically upward from a height of 4 feet with an initial speed of 60 feet per second. The formula $$h=4+60 t-16 t^{2}$$ describes the ball's height above the ground, \(h,\) in feet, \(t\) seconds after it was kicked. Use this formula to solve Exercises \(19-20 .\) What was the ball’s height 2 seconds after it was kicked?

Step-by-Step Solution

Answer

The ball's height 2 seconds after it was kicked is 60 feet.

Step 1: Understand the equation

The equation \(h=4+60t-16t^2\) represents the trajectory of a ball over time. The \(h\) represents the ball's height in feet, and \(t\) represents the time in seconds since the ball was kicked. This kind of equation is an example of a quadratic equation, which often appear in physics when dealing with objects in motion.

Step 2: Plug in the value of time

To find the ball’s height 2 seconds after it was kicked, we need to replace \(t\) in the equation with 2. Doing so, we get: \(h=4+60(2)-16(2^2)\).

Step 3: Solve the equation

After replacing \(t\) with 2 in the previous step, now we just need to compute the result to get the height. The equation solves to \(h=4+(120)-(64)\). Finally, perform the addition and subtraction to get the value of \(h\). This equals to \(60\) feet.

Key Concepts

Trajectory of a Ball

Understanding the trajectory of a ball in motion is a foundational concept in physics. It refers to the path that a ball follows through space as a result of various forces acting upon it. When you kick a football into the air, it makes a curved path; this is what physicists refer to as its trajectory. In the case of the exercise mentioned, the trajectory is described by the equation
\[ h = 4 + 60t - 16t^2 \]
This equation factors in the initial height from which the ball was kicked (4 feet), the initial speed (60 feet per second), and the acceleration due to gravity (-32 ft/s^2, which is often halved in equations to -16 ft/s^2 when solving for height). These elements combine to create a parabolic curve, which is a common shape for the trajectory of projectiles under the influence of gravity. By analyzing this equation and solving it for specific values of time (\( t \)), students can discover the ball’s height at any given moment after it was kicked.

Algebraic Representation of Motion

The algebraic representation of motion simplifies the complex movements of objects into equations that can be solved mathematically. In this example, the motion of a football kicked into the air can be represented by the quadratic equation
\[ h = 4 + 60t - 16t^2 \]
This equation encompasses the initial conditions of the ball’s motion, including its initial height and velocity, and the constant acceleration due to gravity. Using algebra, we can manipulate this equation to find not only the ball's height at any particular time but also other insights, such as how long it takes for the ball to reach the ground, or its maximum height. Algebra serves as a powerful tool for predicting the outcome of moving objects, a method commonly used in various fields of physics and engineering.

Solving Quadratic Equations

Solving quadratic equations is a crucial skill in both mathematics and physics. A quadratic equation is generally of the form
\[ ax^2 + bx + c = 0 \]
where \( a, b, \text{and } c \) are constants and \( x \) is the variable to be solved for. To solve such an equation, one can use various methods such as factoring, completing the square, using the quadratic formula, or graphing. In our physics exercise, when we plug in the value of 2 seconds into the equation of the ball's height
\( h = 4 + 60t - 16t^2 \),
we effectively solve a quadratic equation. Though in this case, it simplifies as we substitute the value of \( t \) and calculate the result, which doesn't require the steps used for more complex quadratics. Understanding these solutions helps students grasp not just the specific problem at hand, but also broader applications involving quadratic equations.

Physics Applications in Algebra

The intersection of physics and algebra is most apparent in problems involving motion, force, and energy. Algebraic equations are used to describe various physical phenomena. In our example, we observe how the quadratic equation models the height of a ball over time, incorporating initial velocity, gravity, and starting height into a coherent algebraic expression. This application is just one of many examples in which algebra serves as a critical tool in physics.
Students should recognize that the principles of algebra underpin essential physics concepts like motion and can be applied to predict the behavior of systems ranging from simple projectiles to complex interactions in astrophysics. Moreover, mastering algebra allows scientists and engineers to create models that solve real-world problems and innovate through new technologies.