Problem 28 - Thomas Calculus

Question

Where trajectories crest For a projectile fired from the ground at launch angle \(\alpha\) with initial speed \(v_{0},\) consider \(\alpha\) as a variable and \(v_{0}\) as a fixed constant. For each \(\alpha, 0<\alpha<\pi / 2,\) we obtain a parabolic trajectory as shown in the accompanying figure. Show that the points in the plane that give the maximum heights of these parabolic trajectories all lie on the ellipse $$ x^{2}+4\left(y-\frac{v_{0}^{2}}{4 g}\right)^{2}=\frac{v_{0}^{4}}{4 g^{2}} $$ where \(x \geq 0\). Graph cannot copy

Step-by-Step Solution

Answer

The points of maximum height lie on an ellipse given by \(x^2 + 4\left(y - \frac{v_0^2}{4g}\right)^2 = \frac{v_0^4}{4g^2}\).

Step 1: Understanding the Problem

We need to find the points where the maximum heights of parabolic trajectories for different launch angles all lie. These points form a curve which is given by an ellipse.

Step 2: Maximum Height of a Projectile

The maximum height achieved by a projectile launched with speed \(v_0\) and angle \(\alpha\) is given by \(y_{max} = \frac{v_0^2 \sin^2(\alpha)}{2g}\). This is calculated using the vertical component of the velocity and the kinematic equation for motion under gravity.

Step 3: Horizontal Position at Maximum Height

At maximum height, the horizontal position \(x\) can be calculated using the horizontal component of the velocity. The position is \(x = \frac{v_0^2 \sin(2\alpha)}{2g}\). Use the equation \(x = v_0 \cos(\alpha) \cdot t\) and the time to reach the maximum height \(t = \frac{v_0 \sin(\alpha)}{g}\).

Step 4: Substituting Trigonometric Identities

Substitute \(\sin^2(\alpha) = \frac{1 - \cos(2\alpha)}{2}\) and \(\sin(2\alpha) = 2 \sin(\alpha) \cos(\alpha)\) into the expressions for \(x\) and \(y_{max}\) to express them only in terms of \(\cos(2\alpha)\).

Step 5: Parametric Equations for the Trajectory

Express the horizontal and vertical coordinates of maximum height with \(\cos(2\alpha)\) as the parameter. Substitute back the trigonometric identities into the equations from step 3 and 4.

Step 6: Eliminating the Parameter

Eliminate the parameter \(\cos(2\alpha)\) to obtain an equation solely in terms of \(x\) and \(y\), which gives the relationship between these coordinates. This requires solving a system of equations and utilizing identity \(1 - \cos^2(2\alpha) = \sin^2(2\alpha)\).

Step 7: Deriving the Equation of the Ellipse

After eliminating the parameter, you'll derive the relation \(x^2 + 4\left(y - \frac{v_0^2}{4g}\right)^2 = \frac{v_0^4}{4g^2}\), confirming the path of maximum height points is an ellipse.

Key Concepts

Parabolic Trajectories

Have you ever noticed that when something is thrown into the air, it often follows a curved path? This specific curved path is called a parabolic trajectory. When a projectile, like a ball, is launched at an angle into the air, the forces acting on it include gravity, which pulls it back down. However, the projectile's initial motion and speed dictate the path, which turns out to be a parabola.

Key features of a parabolic trajectory include:

• The launch angle, represented as \( \alpha \), which is the angle at which the object is shot or thrown.
• The initial speed, or \( v_0 \), which helps determine how far and high the object will travel.
• The effect of gravity, \( g \), which pulls the object downward, giving the path its characteristic shape.

These elements come together to form the familiar arc that most sports enthusiasts have witnessed in real life. Understanding these factors helps in predicting, and even calculating, the trajectory of any projectile.

Maximum Height

The highest point of a projectile's path is known as the maximum height. At this point, the vertical component of the velocity is zero, meaning it reaches its peak before gravity pulls it back down.

To calculate the maximum height \( y_{max} \) of a projectile, we use the formula:
\[y_{max} = \frac{v_0^2 \sin^2(\alpha)}{2g}\]
Here, \( v_0 \sin(\alpha) \) represents the vertical component of the initial velocity, determining how high the projectile goes.

Understanding maximum height is essential, particularly in sports and sciences, as it enables planners and analysts to optimize performance and safety. By knowing the initial parameters, you can adjust them to reach the desired height, predicting whether obstacles can be surpassed.

Ellipse Equation

An ellipse is a beautiful and symmetrical shape, often described as a stretched circle. In the context of projectile motion, we encounter it when describing the maximum height points of various trajectories.

Imagine a series of trajectories launched from the same initial speed but with different angles. The points that represent their highest elevation form an ellipse on a two-dimensional plane. The equation for this ellipse is:
\[x^2 + 4\left(y - \frac{v_0^2}{4g}\right)^2 = \frac{v_0^4}{4g^2}\]
This equation ties together the horizontal distance \( x \), the adjusted vertical height \( y \), and constants deriving from the launch conditions.

• The term \( \frac{v_0^2}{4g} \) adjusts the center of the ellipse vertically.
• The constants \( \frac{v_0^4}{4g^2} \) determine the stretch and shape of the ellipse.

By understanding this equation, we can predict where the highest points will land, how far they'll spread, and their general pattern.

Trigonometric Identities

Trigonometric identities are important tools in mathematics, especially in analyzing projectile motions. The primary trigonometric identities used in calculations involve sine and cosine functions, such as \( \sin(\alpha) \) and \( \cos(\alpha) \).

Key identities include:

• \( \sin^2(\alpha) = \frac{1 - \cos(2\alpha)}{2} \)
• \( \sin(2\alpha) = 2 \sin(\alpha) \cos(\alpha) \)

These identities allow simplification of expressions involving angles and help in eliminating parameters to derive concise solutions.

Through these identities, we can systematically convert complex equations into manageable forms. For instance, elimination of \( \cos(2\alpha) \) simplifies maximum height and horizontal distance calculations, which are crucial for mapping out projectile trajectories.