Problem 30 - Thomas Calculus

Question

Where trajectories erest 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 obtable 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 \(x \geqq 0\)

Step-by-Step Solution

Answer

The maximum heights of the projectile are on the ellipse.

Step 1: Identify the Projectile Motion Equation

For a projectile launched with initial speed \(v_0\) at an angle \(\alpha\), the equation for the maximum height \(y_{max}\) is given by \(y_{max} = \frac{v_0^2 \sin^2 \alpha}{2g}\) where \(g\) is the acceleration due to gravity. The horizontal distance (range) when it reaches maximum height is \(x = \frac{v_0^2 \sin(2\alpha)}{2g}\).

Step 2: Derive Relationship of Maximum Height Points

The equation for any point \( (x, y) \) on the trajectory of the projectile is \( y = x \tan \alpha - \frac{gx^2}{2v_0^2 \cos^2 \alpha} \). At maximum height, this simplifies to \( y_{max} = \frac{v_0^2 \sin^2 \alpha}{2g} \) and \( x= \frac{v_0^2 \sin(2\alpha)}{2g} \).

Step 3: Substitute into Ellipse Equation

Substitute \( x = \frac{v_0^2 \sin(2\alpha)}{2g} \) and \( y = \frac{v_0^2 \sin^2 \alpha}{2g} \) into the ellipse equation \( x^2 + 4\left(y - \frac{v_0^2}{4g}\right)^2 = \frac{v_0^4}{4g^2} \).

Step 4: Verify the Ellipse Equation

After substitution, verify if both sides of the ellipse equation match. Start by calculating \( (y - \frac{v_0^2}{4g}) \) and simplifying, then simplifying the left side of the ellipse equation using trigonometric identities to match the right-hand side \( \frac{v_0^4}{4g^2} \). This ensures the points with maximum heights satisfy the ellipse equation.

Step 5: Conclusion

Since every substitution and simplification shows that the points indeed satisfy the ellipse equation, this proves that for any \( \alpha \), the maximum heights lie on the given ellipse.

Key Concepts

Trajectory

In studying projectile motion, the term **trajectory** refers to the path that an object follows when it is thrown or launched into the air. For a projectile launched at an angle \( \alpha \) with an initial speed \( v_0 \), the trajectory is shaped by both the horizontal and vertical components of the velocity. This means that the projectile will travel in a curved path due to the influence of gravity.
The main characteristics of a projectile's trajectory include:

• It starts at the launch point, where the initial speed breaks down into horizontal and vertical components.
• Gravity affects only the vertical motion, pulling the projectile downward, which forms a concave-up parabola.
• The maximum height and the range of the projectile are determined by both the angle \( \alpha \) and the speed \( v_0 \).

This understanding of trajectory is crucial for tasks where predicting the path of the projectile is necessary, such as in sports, engineering, and simulation scenarios.

Parabolic Motion

In the context of projectile motion, **parabolic motion** refers to the specific type of trajectory that is shaped like a parabola. This occurs because when a projectile is launched at an angle (other than straight up or down), it moves forward in the horizontal direction while simultaneously being accelerated downward due to gravity.
The equation that describes the parabolic motion of a projectile at maximum height is \( y = x \tan \alpha - \frac{gx^2}{2v_0^2 \cos^2 \alpha} \). At maximum height, this simplifies to the expression \( y_{max} = \frac{v_0^2 \sin^2 \alpha}{2g} \).
Key points about parabolic motion include:

• The maximum height of the trajectory depends on \( \sin^2 \alpha \), reflecting how steeply it was launched.
• The horizontal range of the motion when at maximum height is represented by \( x = \frac{v_0^2 \sin(2\alpha)}{2g} \), which shows the dependency on the double angle \( 2\alpha \).
• Each different launch angle below \( 90^{\circ} \) yields a different parabolic path, demonstrating how dynamic these motions can be.

Understanding parabolic motion helps predict where a projectile will land and how high it will go, important for accurate targeting in practical applications.

Ellipse Equation

When exploring the concept of a projectile's maximum height across various angles, the points defining these heights in the plane can actually lie on an **ellipse**. This arises from the geometry of parabolic paths for different launch angles, forming a shape akin to an ellipse when plotted together.
The specific ellipse equation associated with maximum heights can be expressed as:\[x^2 + 4\left(y - \frac{v_0^2}{4g}\right)^2 = \frac{v_0^4}{4g^2}\]where \( x \geq 0 \). This equation represents how the parabolic paths' maximum heights align geometrically.
Key aspects of relating projectile motion to an ellipse include:

• The derivation involves substituting the expressions for the maximum height \( y = \frac{v_0^2 \sin^2 \alpha}{2g} \) and the horizontal component \( x = \frac{v_0^2 \sin(2\alpha)}{2g} \).
• Geometrically, this alignment of maximum heights forms an ellipse, signifying how variations in launch angle can be considered together.
• This unique observation links the otherwise independent parabolic paths into a single coherent structure.

This concept helps students grasp the interaction between different projectile paths revealing how these can connect in unanticipated ways, highlighting the elegance of calculus in action.