Problem 20 - 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 3 seconds after it was kicked?

Step-by-Step Solution

Answer

The ball's height 3 seconds after it was kicked is 40 feet.

Step 1: Substitute the Value for 't'

Substitute the value for 't' with 3 in the given equation, i.e. \( h = 4 + 60(3) - 16(3)^2 \)

Step 2: Calculate Initial Height Plus Upward Motion

Calculate '60t + 4' first as it represents the height gained from the initial kick at 3 seconds, i.e., \( h = 4 + 180 = 184 \) feet

Step 3: Calculate the Height Loss Due to Gravity

Calculate the term '-16t^2' to represent the height loss due to gravity after 3 seconds, i.e., \( h = 184 - 16 * 9 = 184 - 144 = 40 \) feet

Key Concepts

Quadratic Equations

Quadratic equations are fundamental components in algebra that describe a parabola when graphed. They have the standard form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( x \) represents the variable. To solve for \( x \), we can employ different methods, such as factoring, using the quadratic formula, completing the square, or graphing.

In our exercise, the quadratic term \( -16t^2 \) shapes the trajectory of the ball kicked into the air. This term specifies how the ball's height changes over time due to gravity's acceleration. Gravity's pull is represented as a negative effect on the ball's upward motion, which is why the coefficient of the \( t^2 \) term is negative in the equation for height.

Projectile Motion

Projectile motion refers to the movement of an object that is thrown or projected into the air, subject only to the force of gravity and air resistance (which is usually neglected in basic problems). It is a classic example of two-dimensional motion in physics. The motion can be analyzed by breaking it down into horizontal and vertical components; however, in our exercise, the ball is kicked vertically, so we are only concerned with the vertical motion.

Vertical projectile motion is characterized by an upward journey against gravity, reaching a peak height, and then a downward descent. The quadratic equation given in the exercise perfectly models this path, with the ball starting at a height of 4 feet and being subject to Earth’s gravitational pull, which accelerates it downwards. The solution to the problem uses principles of projectile motion to determine the height of the ball at a particular time.

Physics in Algebra

Integrating physics with algebra allows for the modeling and solving of physical problems using algebraic equations. This interdisciplinary approach is particularly evident in the way we express the motion of objects. In our exercise, the formula \( h = 4 + 60t - 16t^2 \) is an algebraic representation of the physics governing the projectile motion of the kicked football.

Each term in the equation has a physical significance: the constant '4' represents the initial height from where the ball was kicked, '60t' denotes the initial upward velocity applied to the football, and \( -16t^2 \) captures the effect of gravitational deceleration over time. Understanding each term's contribution to the ball's motion helps students not only solve the problem but also comprehend the underlying physics illustrated through algebra.