Problem 10 - Algebra 2

Question

BASEBALL A baseball player hits a high pop-up with an initial upward velocity of 30 meter: per second, 1.4 meters above the ground. The height \(h(t)\) of the ball in meters \(t\) seconds after being hit is modeled by \(h(t)=-4.9 t^{2}+30 t+\) 1.4. How long does a player on the opposing team have to get under the ball if he opposing 1.7 meters above the ground? Does your answer seem reasonable? Explain.

Step-by-Step Solution

Answer

The player reacts almost instantly, but this seems unreasonable given human limitations.

Step 1: Understand the problem

We need to determine how long the baseball is in the air before it reaches a height of 1.7 meters. This is modeled by the equation \( h(t) = -4.9t^2 + 30t + 1.4 \).

Step 2: Set up the equation

The problem asks for the time \( t \) when the ball is at a height of 1.7 meters. Therefore, we set up the equation: \(-4.9t^2 + 30t + 1.4 = 1.7\).

Step 3: Isolate the quadratic equation

Subtract 1.7 from both sides to isolate the quadratic expression: \(-4.9t^2 + 30t + 1.4 - 1.7 = 0\). Simplifying this, we get \(-4.9t^2 + 30t - 0.3 = 0\).

Step 4: Use the quadratic formula

The quadratic equation \( ax^2 + bx + c = 0 \) is solved using the formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -4.9 \), \( b = 30 \), and \( c = -0.3 \).

Step 5: Calculate the discriminant

Calculate \( b^2 - 4ac \): \( 30^2 - 4(-4.9)(-0.3) = 900 - 5.88 = 894.12 \).

Step 6: Find the roots

Using the quadratic formula, calculate the roots: \( t = \frac{-30 \pm \sqrt{894.12}}{-9.8} \). First solve for the positive root since it represents time passing.

Step 7: Solve for positive time

Calculate \( \sqrt{894.12} \) which is approximately \( 29.9 \), and then solve for \( t \): \( t = \frac{-30 + 29.9}{-9.8} \), which simplifies to \( t \approx 0.01 \) when using the positive root.

Step 8: Check reasonableness

Given that the player has only 0.01 seconds to react, this suggests the calculation needs re-evaluation or consideration of practical human reaction time is necessary.

Key Concepts

Physics in Algebra

The interplay between physics and algebra can often seem abstract, but it is essential in solving real-world problems. When we look at the motion of a baseball, as in our exercise, we're using algebra to model physical phenomena. Algebra allows us to create an equation that represents the height of the baseball over time, known in physics as the path of a projectile. By setting this equation equal to a specific height—in this case, 1.7 meters, the height of an opposing player—we employ algebra to calculate when the ball will be at that height.

Algebra translates real-world scenarios into mathematical formulas, which can then be manipulated to find solutions. It's crucial to understand both the physics concept, such as gravitational influence on a baseball, and the algebraic method, such as solving quadratic equations, to accurately make these predictions.

• Algebra transforms physical motion into a formula.
• It helps us predict future states of moving objects.
• Combining algebra and physics gives a deeper understanding of the scenario.

Solving Quadratic Equations

Quadratic equations often pop up in physics-related problems, especially those involving projectile motion. The formula for quadratic equations is typically given as \( ax^2 + bx + c = 0 \). Solving these equations can determine critical points of a path, such as the maximum height of a baseball or when the ball reaches a particular height.

The quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is an excellent tool for finding roots. These 'roots' are the solutions for the variable, in this case, time \( t \).

To solve, one must first arrange the equation into standard form, then evaluate each part of the formula. Understanding the discriminant \( b^2 - 4ac \) helps predict the number and type of solutions without complete calculation.

• The quadratic formula provides reliable solutions for parabolic paths.
• Understanding each part of the formula simplifies problem-solving.
• Practice with quadratic equations enhances accuracy and confidence.

Projectile Motion

Projectile motion is a type of problem often modeled by quadratic equations due to the parabolic path the objects take under uniform gravity. When a baseball is hit into the air, its path is shaped both by the initial velocity given to it and the gravitational pull acting on it throughout its flight. Importantly, this vertical motion is independent of horizontal motion unless otherwise specified.

A projectile's motion is usually described using two components: its upward or downward velocity, influenced by gravity, and its horizontal motion, often assumed constant if air resistance is ignored. In our exercise, the equation showing the height of the baseball is a result of solving for the gravitational influence combined with the initial upward velocity and starting height.

• Projectile motion is commonplace in sports like baseball.
• It uses the principles of gravity and initial velocity.
• Understanding both vertical and horizontal movements is critical.