Question

Describe a real-life story that could be modeled by \(h=-16 t^2+v_0 t+h_0\). Write the height model for your story and determine how long your object is in the air.

Short answer

In our real-life story of a football being kicked upwards from a goal post, the football follows the height model \(h=-16t^2 + 64t + 4\). The football remains in the air for a total time of approximately \(4.127\) seconds.

Step-by-step solution

Scenario Creation

Let's consider a scenario where a football is kicked upwards from the goal post. The football reaches a peak and then descends due to gravity. The goal post is \(4\) feet high, the initial velocity of the football kicked is \(64\) feet per second. So, the height model would be \(h=-16t^2 + 64t + 4\).

Find the Time of Flight

To find out how long the ball is in the air (time of flight), we need to find when the height \(h\) of the ball is \(0\). This means the ball has returned to the ground. We need to solve the quadratic equation \( -16t^2 + 64t + 4 = 0\). We can simplify this to \( t^2 - 4t - 0.25 =0\).

Solve the Quadratic Equation

To solve the quadratic equation \( t^2 - 4t - 0.25 =0 \), we can use the quadratic formula \( t= \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \(a=1\), \(b=-4\), and \(c=-0.25\). Substituting these values into the quadratic formula gives \( t= 4.127s \) or \( t= -0.127s \). Since time cannot be negative, the time of flight is \( t= 4.127s \).

Key concepts

Projectile Motion

Projectile motion is a fascinating concept that describes the path an object follows when it is thrown or launched into the air and is influenced by gravity. It applies to a wide range of real-world scenarios, from sports to physics experiments.
When examining projectile motion, one often encounters a parabolic trajectory. This is due to the consistent force of gravity acting on the object, pulling it downward.
To mathematically model this motion, we often use a quadratic equation. The standard equation to describe the height of an object in projectile motion is: \[ h = -16t^2 + v_0t + h_0 \] where:

• \(h\) is the height at time \(t\),
• \(t\) is the time in seconds,
• \(v_0\) is the initial velocity,
• \(h_0\) is the initial height from which the object is launched.

Understanding this formula allows us to predict where and when the object will reach its peak or return to the ground.

Time of Flight

The time of flight is the duration for which a projectile remains in the air before hitting the ground. It's an important consideration in analyzing projectile motion because it affects where and how a projectile lands.
To determine the time of flight, one calculates the time it takes for the projectile's height to become zero again, meaning it has returned to ground level. This is done by solving the height equation \( h = 0 \).
For instance, if a football is initially kicked with an upward velocity, it rises until gravity slows it down, stops it momentarily, and then pulls it back down. Solving the quadratic equation for \(t\) when \(h = 0\) gives us the time of flight. This exercise involves finding the roots of the equation, often resulting in two possible times:

• One corresponding to the time at launch.
• The other indicating when it returns to the ground.

Since time cannot be negative, we only consider the positive root, which represents the actual time of flight.

Initial Velocity

The initial velocity \(v_0\) plays a crucial role in determining how far and high a projectile can travel. It refers to the speed at which the object is launched at the moment it begins its motion. Multiple factors, such as speed and angle of launch, influence the initial velocity.
When we input an initial velocity in our height equation, it helps calculate the projectile's behavior:

• The higher the initial velocity, the longer the time of flight.
• Larger initial velocities also result in higher maximum heights.
• It directly impacts the distance covered horizontally.

In our example, the initial velocity is given as 64 feet per second, significantly affecting the flight duration and peak height of the kicked football. Understanding initial velocity ensures accurate predictions of projectile motion outcomes.

Solving Quadratic Equations

Solving quadratic equations is a fundamental mathematical skill necessary for analyzing projectile motions. Quadratic equations like \(-16t^2 + 64t + 4 = 0\) emerge naturally in these scenarios, requiring specific techniques to find solutions.
The equation's roots represent critical points in the projectile's path, such as the start and impact times.
There are several methods to solve quadratic equations:

• Factoring (if possible), yielding exact roots quickly.
• The quadratic formula: \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This is universally applicable and always provides solutions, even when factoring is impossible.
• Completing the square, useful for visualizing solutions but less efficient for quick calculations.

In our step-by-step solution, the quadratic formula was used, yielding two solutions. Selecting the positive value gives us meaningful results for time in our problem, highlighting the solution's practical utility.