Problem 36 - Algebra 2

Question

PHYSICS For Exercises \(34-36,\) use the following information. An object is fired straight up from the top of a 200 -foot tower at a velocity of 80 feet per second. The height \(h(t)\) of the object \(t\) seconds after firing is given by \(h(t)=-16 t^{2}+80 t+200\) Interpret the meaning of the \(y\) -intercept in the context of this problem.

Step-by-Step Solution

Answer

The y-intercept is 200, representing the tower's height.

Step 1: Identify the Linear Equation

The given function is a quadratic equation of the form \(h(t) = -16t^2 + 80t + 200\). This equation models the height of the object above the ground at any time \(t\).

Step 2: Determine the y-intercept

The y-intercept of a quadratic equation \(h(t) = at^2 + bt + c\) is the value of the function when \(t = 0\). This means we need to find \(h(0)\).

Step 3: Calculate the y-intercept Value

Substitute \(t = 0\) into the equation: \[ h(0) = -16(0)^2 + 80(0) + 200 = 200 \]Thus, the y-intercept is 200.

Step 4: Interpret the y-intercept

The y-intercept represents the initial height of the object at the moment it is fired. Since the object is fired from the top of a 200-foot tower, the y-intercept indicates that the initial height is 200 feet.

Key Concepts

Understanding the Y-Intercept in Quadratic Equations

When examining quadratic equations, such as the equation for the height \( h(t) = -16t^2 + 80t + 200 \), we often look at the y-intercept to understand initial conditions. The y-intercept is the point where the graph of the equation crosses the y-axis, which occurs when \( t = 0 \). In this specific context, it tells us the height of the object right at the beginning of the scenario.
In our given problem, when the object is fired, it is already sitting 200 feet above the ground on top of the tower, which is the y-intercept of the equation. By substituting \( t = 0 \) into the equation, we find that \( h(0) = 200 \). Hence, the y-intercept of 200 feet represents the initial height of the object before any time has passed.
Understanding this helps you see how tall the object is relative to the ground before it begins its ascension under the influence of its initial velocity and gravity. It's a crucial starting point to consider before analyzing further aspects like how high the object will go or when it will return to the ground.

Exploring Parabolic Motion

Parabolic motion is a type of motion experienced by a projectile moving under the influence of gravity. The path followed by such an object is described as a parabola. This is why the equation for our problem is quadratic, involving a term with \( t^2 \).
The shape of the parabola is determined by the coefficients of the equation. In \( h(t) = -16t^2 + 80t + 200 \), the "-16" is especially important. It represents acceleration due to gravity, indicating that the object will curve back downwards after reaching its peak height. The coefficient "-16" comes from physics, modeling gravitational pull acting on bodies even if they are thrown straight up.
The parabola has a vertex at the highest point of the object's journey. From launch, the object travels upwards, slows under gravity, reaches a maximum height and begins to descend. Analyzing parabolic motion involves understanding how the object behaves at different time intervals through its entire trajectory.

Unveiling Initial Velocity

Initial velocity is crucial in projectile motion as it determines how fast and how far an object travels after being launched. In the equation \( h(t) = -16t^2 + 80t + 200 \), the term with 't' represents velocity, specifically initial velocity, which in this problem is 80 feet per second. This value affects both the maximum height the object will achieve and the overall time it will stay in the air.
Initial velocity is the speed and direction the object has at the moment it's launched. It's typically positive when an object is fired upwards. In our equation, 80 feet per second means that right after being launched, the object travels at that speed towards the peak of its trajectory.
To visualize this, think of initial velocity as the power behind a toss. A stronger initial velocity means the object will go higher and further before gravity causes it to come back down. In problems like the one given, calculating how initial velocity interacts with gravity helps predict other important motion features like maximum height and time until landing.