Problem 48 - Algebra and Trigonometry with Analytic Geometry

Question

In the \(1940 \mathrm{~s}\), the human cannonball stunt was performed regularly by Emmanuel Zacchini for The Ringling Brothers and Barnum \& Bailey Circus. The tip of the cannon rose 15 feet off the ground, and the total horizontal distance traveled was 175 feet. When the cannon is aimed at an angle of \(45^{\circ}\), an equation of the parabolic flight (see the figure) has the form \(y=a x^{2}+x+c\). (a) Use the given information to find an equation of the flight. (b) Find the maximum height attained by the human cannonball.

Step-by-Step Solution

Answer

(a) The equation is \(y = -0.0057x^2 + x + 15\). (b) Maximum height is about 95.22 feet.

Step 1: Establish known variables

We know from the problem that the flight path of the human cannonball is at a 45-degree angle with a total horizontal distance, or range, of 175 feet. This range is the distance across the x-axis. The tip of the cannon is 15 feet off the ground, which is a point on the parabola: \((0, 15)\).

Step 2: Understand the parabolic equation

The standard form of a quadratic equation for a parabola is given by: \(y = ax^2 + bx + c\).In this form, we have two unknowns: \(a\) and \(c\). We know that the starting point of the cannon is 15 feet off the ground when \(x=0\), so \(c=15\).

Step 3: Use horizontal range to solve for 'a'

When the cannon reaches the end of its horizontal range (175 feet), it also returns to the height it was launched from (15 feet). We can use the equation \(a(175)^2 + (175) + 15 = 15\) to solve for \(a\). This simplifies to:\[a(175)^2 + 175 = 0\]\[30625a + 175 = 0\]\[30625a = -175\]\[a = -\frac{175}{30625}\].

Step 4: Solve for 'a'

Calculate the value of \(a\) by simplifying the fraction \[a = -\frac{175}{30625} \approx -0.0057\].This gives us the complete equation of the flight as \(y = -0.0057x^2 + x + 15\).

Step 5: Vertex form to find maximum height

The parabolic flight is symmetric, and the maximum height is at the vertex of the parabola. For a parabolic equation \(y = ax^2 + bx + c\), the x-coordinate of the vertex is given by \(x = -\frac{b}{2a}\).Substitute \(b=1\) and \(a=-0.0057\): \[x = \frac{-1}{2(-0.0057)} \approx 87.72\].

Step 6: Calculate the maximum height

Substitute the x-value of the vertex back into the parabolic equation to find the maximum height: \[y = -0.0057(87.72)^2 + 87.72 + 15\].Compute the value:\[-0.0057(7700.5584) + 87.72 + 15 \approx 95.22\]. Thus, the maximum height attained by the human cannonball is approximately 95.22 feet.

Key Concepts

Parabolic Flight

In the context of projectile motion, parabolic flight is a type of motion where an object follows a curved path called a parabola. This happens when an object is thrown, launched, or projected near the earth's surface and moves under the influence of gravitational acceleration only. Whether it is a human cannonball or a basketball shot, the path can be modeled using a parabolic equation. A few things characterize parabolic flight:

• The motion is two-dimensional.
• It can be described using a quadratic equation.
• The trajectory is symmetric around its highest point, known as the vertex.

Let's explore how the understanding of this concept helps us solve problems involving projectile motion, such as finding the maximum height or the range of a projectile.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form: \[ y = ax^2 + bx + c \] where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable. This equation describes a parabola in the cartesian plane. In our exercise, the parabolic flight of the cannonball is best represented by a quadratic equation, allowing us to express relationships between distance and height:

• The coefficient \(a\) determines the shape of the parabola. If \(a\) is negative, the parabola opens downwards; if positive, it opens upwards.
• The term \(b\) influences the slope.
• The constant \(c\) is the y-intercept, indicating where the parabola crosses the y-axis.

Understanding quadratic equations allows us to analyze and predict a projectile's path using mathematical concepts to derive valuable information like velocity, distance, and height.

Maximum Height

The maximum height of a projectile in parabolic motion is achieved at the vertex of the parabola. This is crucial because it represents the highest point the projectile reaches. In the problem, finding the maximum height involves determining the vertex of the quadratic trajectory equation. The maximum height can be calculated using:

• The vertex formula \(x = -\frac{b}{2a}\) gives the x-coordinate at which the maximum height occurs.
• Substituting this \(x\)-value back into the parabolic equation determines the maximum height \(y\).

For the cannonball problem, once we solve for this key point on the x-axis (across the horizontal distance), we understand how high off the ground the human cannonball is at the peak of their hyperbolic flight. Knowing the maximum height is essential not only for solving theoretical problems but also for practical safety and design considerations in real-world applications like stunts or sports.

Vertex of a Parabola

The vertex of a parabola is its highest or lowest point, depending on whether the parabola opens upwards or downwards. In our scenario, the vertex is the highest point, representing the maximum height of the cannonball. To find the vertex, we use the standard formula:
\[x = -\frac{b}{2a}\] and substitute this \(x\)-value back into the equation to solve for \(y\), giving the vertex coordinates \((x, y)\).
For a downward opening parabola like the one in the cannonball problem:

• The x-value of the vertex tells us how far the projectile travels horizontally before reaching the maximum height.
• The y-value gives us the peak height, crucial for understanding and optimizing the trajectory.

The vertex is a fundamental concept in parabolas as it represents the point of symmetry and maximum value in contexts like projectile motion, maximizing our understanding of the parabola's characteristics in equations.