Question

A lacrosse player throws a ball in the air from an initial height of 7 feet. The ball has an initial vertical velocity of 90 feet per second. Another player catches the ball when it is 3 feet above the ground. How long is the ball in the air?

Short answer

The time the ball is in the air and thus caught is approximately 2.8 s or 5.7 s. However, considering context, we realize that the ball can't take 5.7 s to reach a height of 3 feet, if it was initially thrown with a velocity of 90 ft/s. So, we discard this solution. Therefore, the ball is in the air for about 2.8 seconds before it is caught.

Step-by-step solution

Set Up the Equation

Prominent equation to be used here is the second equation of kinematic motion which is \( h = v_i t - 0.5 g t^2 \). Substituting the given values, we get: \( 3 = 90t - 0.5 * 32.2 * t^2 \). This is a quadratic equation in standard form.

Normalizing the Equation

In order to simplify the equation, we can first multiply the entire equation by 2 to get rid of the 1/2 which will yield \( 6 = 180t - 32.2t^2 \). Then, rearrange the equation so it's in the standard quadratic form \( 0 = at^2 + bt + c. \) Hence, we obtain \( 0 = -32.2t^2 + 180t - 6 \).

Solve the Quadratic Equation

We utilize the quadratic formula, \( t = (-b ± sqrt(b^2 - 4ac)) / (2a) \), to find \( t \). Substituting \( a = -32.2 \), \( b = 180 \), \( c = -6 \) into the equation will give us two solutions.

Find the Valid Solution

Among the two solutions from the quadratic formula, only one will make sense in our context (time cannot be negative). The valid answer will be the positive value.

Key concepts

Kinematic Motion

Kinematic motion deals with the movement of objects and the forces that cause this movement. It involves different equations that describe an object's velocity, acceleration, time, and displacement. In this example, we use the kinematic equation:

• \( h = v_i t - 0.5 g t^2 \)

This equation helps us determine how long a lacrosse ball, thrown by a player, stays in the air. Here, \( h \) represents the height above the ground. \( v_i \) is the initial vertical velocity, \( g \) is the acceleration due to gravity (approximately 32.2 feet per second squared on Earth), and \( t \) is the time in seconds.
We know the initial height is 7 feet, and the final height is when another player catches the ball at 3 feet. By substituting these values, this kinematic equation turns into a quadratic equation that we will learn to solve.

Quadratic Formula

Quadratic equations arise frequently in kinematic motion problems. In this exercise, after substituting the known values into the kinematic equation, we are left with:

• \( 0 = -32.2t^2 + 180t - 6 \)

which is in the standard form of a quadratic equation \( 0 = at^2 + bt + c \).
To solve for \( t \), we use the quadratic formula:

• \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

"The terms \( a \), \( b \), and \( c \) are the coefficients of the equation, where \( a = -32.2 \), \( b = 180 \), and \( c = -6 \). This formula will give us two solutions, since quadratic equations can have two roots. However, not all solutions are applicable in our physical context of time.

Problem Solving

In solving physics problems, particularly those involving kinematic motion, it's critical to follow a structured approach:

• Identify the knowns and unknowns: In our example, we know the initial vertical velocity, the initial and final heights, and the acceleration due to gravity. We need to find the time \( t \).
• Write the given equation: Use relevant kinematic equations to describe the motion, here it's a quadratic equation.
• Manipulate the equation: Convert it to standard quadratic form to utilize the quadratic formula.
• Solve for the unknowns: Carefully apply the quadratic formula, substituting correct values.
• Validate the answer: Ensure the solutions make sense—time should be positive.

This structured approach not only helps solve kinematic problems but also enhances overall problem-solving skills.

Initial Velocity

Initial velocity is a key factor in determining the motion of a moving object. In this problem, the ball's initial velocity is 90 feet per second.
The initial velocity determines how quickly and in what direction the ball will move at the start. It's part of the kinematic equation:

• \( h = v_i t - 0.5 g t^2 \)

where \( v_i \) is the initial velocity.
It significantly affects how long the ball will stay in the air before being caught. A higher initial velocity increases the time of flight.
Understanding initial velocity is crucial for predicting the behavior of any projectile, whether in sports like lacrosse or other real-world applications.