Problem 19 - College Algebra

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 formula

The formula provided in the problem, \(h = 4 + 60t - 16t^2\) describes the ball's height above the ground over time. The time, \(t\), is in seconds and the height, \(h\), is in feet.

Step 2: Substituting the time

The time given in the problem is 2 seconds. Substituting \(t = 2\) into the formula gives \(h = 4 + 60*2 - 16*2^2\).

Step 3: Calculating the height

Calculate the value by following the operator's precedence (multiply and square first, then execute addition and subtraction): \(h = 4 + 120 - 64 = 60\)

Key Concepts

Projectile Motion

Projectile motion describes the movement of an object that is thrown, kicked, or launched into the air and moves under the influence of gravity alone. It is important to understand that the motion has both horizontal and vertical components, which are independent of each other. However, in the case of an object kicked straight up, like the football in our example, there is no horizontal movement, and we only consider the vertical motion.

When an object is kicked vertically upward, it will rise until the force of gravity slows it down to a temporary stop at its peak height, before it starts to fall back to the ground. This up-and-down movement can be described by a quadratic equation, which in this case tells us how the height of the football changes with time.

Quadratic Equation

A quadratic equation is a second-degree polynomial, typically written in the form of \( ax^2 + bx + c = 0 \). In our football example, the equation \(h = 4 + 60t - 16t^2\) shows the characteristics of a quadratic equation where the position of the football at any given time, \(h\), depends on the time squared, \(t^2\), and other factors affecting its motion. This equation can be used to determine the maximum height of the football, the time it takes to reach the ground, or its height at a specific time interval, as seen in the exercise.

By re-arranging this type of equation, one can solve for the value of \(t\) when \(h\) is known, or vice versa, by using various techniques to solve quadratic equations.

Time-Height Relationship

The time-height relationship in projectile motion describes how the height of the projectile changes over time. In our exercise, this relationship is represented by the quadratic equation \(h = 4 + 60t - 16t^2\). By inserting a specific time into the equation, we can calculate the corresponding height. This is a crucial step for solving problems in projectile motion because it provides a direct connection between time and position.

Understanding this relationship helps in predicting how high the projectile will go and how long it will take to reach a certain height, or to return to the ground. It’s part of the analysis that defines the trajectory of the projectile.

Solving Quadratic Problems

Solving quadratic problems involves finding the values of the variable that satisfy the quadratic equation. In this case, we are given a specific time (2 seconds) and we need to find the height of the football. By substituting the given time into our equation, we perform operations in a specific order—multiplication and squaring before addition and subtraction—to solve for height.

To effectively solve quadratic problems, students should be familiar with strategies like factoring, completing the square, or using the quadratic formula. However, when given specific values to substitute as we have here, solving can be more straightforward. Practice and understanding the operator's precedence can significantly simplify the process.