Problem 63 - Algebra and Trigonometry

Question

Height of a Ball If a ball is thrown directly upward with a velocity of 40 \(\mathrm{ft} / \mathrm{s}\) , its height (in feet) after \(t\) seconds is given by \(y=40 t-16 t^{2} .\) What is the maximum height attained by the ball?

Step-by-Step Solution

Answer

The maximum height attained by the ball is 25 feet.

Step 1: Identify the equation

We are given the height equation: \( y = 40t - 16t^2 \), where \( y \) is the height in feet of the ball after \( t \) seconds.

Step 2: Recognize the type of equation

The equation \( y = 40t - 16t^2 \) is a quadratic equation in standard form \( y = at^2 + bt + c \) with \( a = -16 \), \( b = 40 \), and \( c = 0 \). This is a parabola that opens downward.

Step 3: Calculate the time at maximum height

To find when the ball reaches its maximum height, use the vertex formula \( t = -\frac{b}{2a} \). Here, \( a = -16 \) and \( b = 40 \). So, \( t = -\frac{40}{2(-16)} = \frac{40}{32} = 1.25 \) seconds.

Step 4: Substitute time into the height equation

Substitute \( t = 1.25 \) back into the height equation to find the maximum height: \( y = 40(1.25) - 16(1.25)^2 \).

Step 5: Solve for the maximum height

Calculate \( y = 40(1.25) - 16(1.25)^2 \). This simplifies to \( y = 50 - 25 = 25 \). Hence, the maximum height attained by the ball is 25 feet.

Key Concepts

Vertex Formula

The vertex formula is crucial in understanding the behavior of quadractic equations. It's applied to find the vertex of a parabola. In mathematical terms, the vertex of a parabola described by the equation \[ y = ax^2 + bx + c \] can be identified via the formula:\[ t = -\frac{b}{2a} \] where:

• \( a \) represents the coefficient of the squared term.
• \( b \) is the coefficient of the linear term.
• \( c \) is the constant term.

This formula reveals when the maximum or minimum point of the parabola is reached.
In the case of a downward-opening parabola, such as ours with a ball thrown upward, the vertex formula pinpoints the moment when the ball is at its maximum height.
By substituting the given values—for our ball example, \( a = -16 \) and \( b = 40 \)—into the formula, we calculate that the vertex occurs at 1.25 seconds.This moment signifies the peak of the ball's journey before it descends back.

Parabolic Motion

Parabolic motion characterizes the path of objects under the influence of gravity when projected through the air. This path is a parabola, a symmetric curve unique to quadratic functions.When a ball is thrown directly upward, as in our exercise, it follows a vertical parabolic path and comes down along the same trajectory.
Key features of parabolic motion include:

• The initial upward velocity imparts the upward arc.
• Gravitational force causes the ball's velocity to decrease as it ascends.
• At the peak, the upward velocity becomes zero before reversing direction.

In this context, the quadratic equation of height, \( y = 40t - 16t^2 \), accurately models the parabolic trajectory. Here:- The coefficient \( 40 \) indicates the initial velocity in feet per second.- The negative coefficient \(-16\) reflects the gravitational acceleration, presenting the downward pull as the ball moves upward.
This interplay of velocity and gravity creates the distinct parabolic motion and determines that the ball reaches its highest point when momentum is fully countered by gravity.

Maximum Height of a Projectile

The maximum height of a projectile defines the apex of its trajectory, the point at which its vertical velocity turns zero before it starts descending. Identifying this point is crucial for problems involving free-fall motion and can be effectively determined using the vertex formula.
Steps to find maximum height:

• Recognize the quadratic nature of the motion, as signified by equations like \( y = 40t - 16t^2 \).
• Use the vertex formula to identify the time at which maximum height is achieved, \( t = 1.25 \) in our example.
• Substitute this time back into the height equation to find \( y_{max} \). For our projectile, this becomes \( y = 40(1.25) - 16(1.25)^2 \), resulting in a height of 25 feet.

This maximum height represents the utmost elevation the projectile reaches under the given initial conditions, such as the upward velocity and gravitational pull. Understanding this concept allows us to predict and analyze the peak performance of projectiles in various contexts.