Problem 102 - College Algebra with Modeling and Visualization

Question

A baseball is thrown downward with an initial velocity of 30 feet per second from a stadium seat that is 80 feet above the ground. Estimate to the nearest tenth of a second how long it takes for the baseball to strike the ground.

Step-by-Step Solution

Answer

The baseball hits the ground after approximately 3.0 seconds.

Step 1: Identify the Given Parameters

We are given an initial velocity of 30 feet per second and an initial height of 80 feet. In this scenario, the motion is under the influence of gravity, which we take to be approximately 32 feet per second squared downward.

Step 2: Use the Equation of Motion

The equation of motion for an object under constant acceleration is given by \( s(t) = s_0 + v_0 t + \frac{1}{2} a t^2 \), where \( s_0 \) is the initial position, \( v_0 \) is the initial velocity, and \( a \) is the acceleration due to gravity.

Step 3: Substitute the Given Values

Given \( s_0 = 80 \) feet, \( v_0 = 30 \) feet/second, and \( a = -32 \) feet/second squared, substitute these into the equation: \( 0 = 80 + 30t - 16t^2 \).

Step 4: Set Up the Quadratic Equation

Rearrange the terms to form a quadratic equation: \( -16t^2 + 30t + 80 = 0 \).

Step 5: Apply the Quadratic Formula

Use the quadratic formula, \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -16 \), \( b = 30 \), and \( c = 80 \), to find the values of \( t \).

Step 6: Calculate the Discriminant

Calculate the discriminant \( b^2 - 4ac \): \( 30^2 - 4(-16)(80) = 900 + 5120 = 6020 \).

Step 7: Determine the Roots

Substitute the discriminant back into the quadratic formula to find the roots: \( t = \frac{-30 \pm \sqrt{6020}}{-32} \). Using only the positive root gives the time it takes to hit the ground.

Step 8: Solve for Time When the Ball Hits the Ground

Calculate \( \sqrt{6020} \), then find \( t = \frac{-30 + \sqrt{6020}}{-32} \). Perform the calculations to get the value of \( t \). The root is approximately 3.0 seconds.

Key Concepts

Understanding Kinematics

To fully grasp how a baseball travels through the air, it's essential to understand the basics of kinematics. Kinematics is the branch of mechanics that describes the motion of objects without necessarily considering the forces that cause this motion. It's mainly concerned with parameters like displacement, velocity, and acceleration.

Kinematic equations allow us to predict the future position of an object based on its initial conditions and influenced by forces like gravity.

These equations are fundamental in solving problems involving projectiles, free fall, and motion in one and two dimensions.

• Displacement refers to how much an object has moved from its original position.
• Velocity is the rate of change of the position of an object, including both speed and direction.
• Acceleration is how much the velocity of an object changes over time.

Kinematic equations simplify complex real-world scenarios into solvable mathematical problems.

Role of Gravity in Motion

Gravity is one of the key forces influencing the motion of objects like the baseball in our example. It is a natural phenomenon by which all things with mass are brought towards one another, including objects towards the Earth. In kinematic equations related to vertical motion, gravity is usually the constant acceleration acting on the object.

In our specific scenario, gravity provides a constant downward acceleration of approximately 32 feet per second squared. This negative acceleration is integral in determining how quickly the baseball will reach the ground.

• The acceleration due to gravity affects the time it takes for an object to fall.
• It causes objects in free fall to speed up as they move downward.

Essentially, gravity ensures that regardless of the initial velocity of an object, it will gradually increase in downward speed until it hits the ground.

Significance of Initial Velocity

When solving motion problems like projecting a baseball downward, the initial velocity plays a pivotal role. Initial velocity is the speed at which an object begins its movement along a path.

In our problem, the baseball is thrown downward with an initial velocity of 30 feet per second. This initial push alters the time it will take for the ball to reach the ground because it provides an initial boost.

• A higher initial velocity means the object reaches the ground faster.
• The initial velocity contributes to the overall motion analysis along with gravitational acceleration.

Understanding how initial velocity fits into the kinematic equation helps learners predict the path and duration of the object's travel precisely.