Problem 42 - Thomas Calculus in SI Units

Question

The range \(R\) of a projectile fired from the origin over horizontal ground is the distance from the origin to the point of impact. If the projectile is fired with an initial velocity \(v_{0}\) at an angle \(\alpha\) with the horizontal, then in Chapter 13 we find that $$R=\frac{v_{0}^{2}}{g} \sin 2 \alpha$$ where \(g\) is the downward acceleration due to gravity. Find the angle \(\alpha\) for which the range \(R\) is the largest possible.

Step-by-Step Solution

Answer

The angle \( \alpha \) for maximum range is 45°.

Step 1: Understanding the Function

The range of the projectile is given by the equation \( R = \frac{v_0^2}{g} \sin 2\alpha \). To maximize the range \( R \), we need to maximize the term \( \sin 2\alpha \) since \( \frac{v_0^2}{g} \) is a constant.

Step 2: Maximizing the Sine Function

The sine function, \( \sin \theta \), achieves its maximum value of 1 when \( \theta = 90^{\circ} \) or \( \theta = \frac{\pi}{2} \) radians.

Step 3: Applying the Condition to Maximize Range

We set \( 2\alpha = 90^{\circ} \) to maximize the range \( R \). Solving for \( \alpha \), we divide both sides by 2, yielding \( \alpha = 45^{\circ} \).

Step 4: Final Check and Conclusion

The maximum value of \( \sin 2\alpha \) is 1, which occurs when \( 2\alpha = 90^{\circ} \). Thus, the range is maximized when \( \alpha = 45^{\circ} \).

Key Concepts

Maximum Range of Projectile

The maximum range of a projectile is a fascinating aspect of physics, focusing on achieving the furthest possible distance from the starting point. The concept primarily depends on the angle and speed at which a projectile is launched. In our scenario, we calculate this using the given formula for range: \[ R = \frac{v_0^2}{g} \sin 2\alpha \] Here, the variables include:

• \(v_0\): the initial velocity of the projectile
• \(g\): the acceleration due to gravity
• \(\alpha\): the angle of projection

Understanding how to maximize this range requires optimizing the projectile's trajectory. Since gravity and initial velocity are constants in this scenario, the range is maximized by controlling \(\alpha\). The goal is to make \(\sin 2\alpha\) as large as possible, reaching its maximum when the proper conditions are set for \(\alpha\).

Angle of Projection

The angle of projection, \(\alpha\), is pivotal in determining how far the projectile will travel. In projectile motion, varying \(\alpha\) affects the vertical and horizontal components of motion differently. Optimizing this angle involves achieving a balance where both components contribute effectively to the distance traveled. To find the angle resulting in the maximum range, we focus on making \(2\alpha = 90^\circ\). This transformation arises because the sine function within our range equation achieves its peak at this point. By setting \(2\alpha = 90^\circ\), we divide both sides by 2, leading to the critical angle:
\[ \alpha = 45^\circ \]A 45-degree angle achieves an ideal state where the initial velocity's energy is equally allocated to vertical lift and horizontal push. This allocation allows the projectile to reach its longest possible travel path while maintaining altitude. Consequently, a 45-degree launch provides the most efficient trajectory for maximizing the distance covered.

Sine Function Optimization

The sine function is integral to calculating projectile range, particularly when using angle adjustments as a variable for maximizing distance. The sine function, expressed in the form \(\sin \theta\), reaches its highest output of 1 when the angle \(\theta\) is 90 degrees (or \(\frac{\pi}{2}\) radians). In our scenario, the necessity to maximize \(\sin 2\alpha\) means that we set \(2\alpha = 90^\circ\), ensuring the sine function reaches this peak value. Achieving this peak is crucial because multiple values can influence the maximum range attainable, yet only those maintaining the 90-degree condition allow \(\sin 2\alpha\) to exhibit its full potential. Understanding this property and applying it to our equation means noticing that any shift away from \(90^\circ\) diminishes sine's contribution. This sensitivity underscores its role as a pivotal point in mathematical and physical analysis of projectile motion. By marrying the angle of projection with sine's properties, a precise and optimized trajectory is achieved, making the sine function's optimization both a mathematical and practical tool in real-world applications.