Problem 78 - Algebra 1: Concepts and Skills

Question

Use a vertical motion model to find how long it will take for the object to reach the ground. Round your solution to the nearest tenth. You throw a ball downward with an initial velocity of -10 feet per second out of a window to a friend 20 feet below. Your friend does not catch the ball.

Step-by-Step Solution

Answer

The exact value of \( t \) can only be reached by calculating the quadratic formula with the given values. The result then has to be rounded to the nearest tenth.

Step 1: Set up the Equation

The equation is given by \( 0 = 0.5*(-32)t^2 -10t -20 \)

Step 2: Simplify the Equation

Upon simplification, the equation becomes \( 0 = -16t^2 -10t -20 \)

Step 3: Reorganize the Equation

For ease in solving, the equation can be reorganized as \( 16t^2 + 10t + 20 = 0 \)

Step 4: Solve the Equation for t using Quadratic Formula

Using the quadratic formula \( t = [-b ± sqrt(b^2 - 4ac)] / (2a) \), we plug in \( a=16 \), \( b=10 \), and \( c=20 \) which yields \( t1 = (-10 - √(100 - 4*16*20)) / 32 \) and \( t2 = (-10 + √(100 - 4*16*20)) / 32 \)

Step 5: Discard Non-Physical Result

The results will yield one positive and one negative value for \( t \). Since time cannot be negative in this physical context, we discard the negative result.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool that provides the solution to quadratic equations of the form \( ax^2 + bx + c = 0 \). In the formula \( x = [-b \pm \sqrt{b^2 - 4ac}] / (2a) \), the symbols \(a\), \(b\), and \(c\) represent the coefficients of the equation's terms, where \(a\) is the coefficient of the \(x^2\) term, \(b\) is the coefficient of the \(x\) term, and \(c\) is the constant term.

When applying the quadratic formula, it's crucial to first ensure that the equation is in its standard form and that you have correctly identified the coefficients. Once the formula is applied, it will typically produce two solutions due to the \(\pm\) sign in the numerator. These solutions could be real or complex numbers. In real-world applications, such as vertical motion problems, only the physical solutions, which are the real numbers that make sense in the context, are taken into consideration.

Initial Velocity

Initial velocity is the velocity at which an object begins its motion. It is an essential component in the equations of motion, especially in vertical motion models where gravity is a force to consider. When an object is thrown, projected, or falls, its initial velocity will influence how the object moves through space over time.

In the context of our exercise, the initial velocity is given as \( -10 \) feet per second, which means that the ball is thrown downwards. The negative sign indicates the direction of the velocity with respect to the chosen coordinate system, where typically upward is positive and downward is negative. Understanding the initial velocity aids in setting up the equation correctly and is indispensable for predicting the object's future position and motion.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, generally in the form \( ax^2 + bx + c = 0 \), where \(a \eq 0\). These equations are capable of describing various physical phenomena, including the motion of objects under constant acceleration, like gravity.

In solving quadratic equations, one can either factor the equation, complete the square, use a graphing method, or apply the quadratic formula, which is often the most direct approach. The exercise uses a vertical motion model, yielding a quadratic equation that describes the height of the object as a function of time. Setting this equation to zero allows for the determination of when the object will reach the ground. Recognizing the structure of quadratic equations and knowing how to manipulate and solve them is fundamental for a wide array of problems in algebra and physics.