91Ó°ÊÓ

Throwing events in track and field include the shot put, the discus throw, the hammer throw, and the javelin throw. The distance that the athlete can achieve depends on the initial speed of the object thrown and the angle above the horizontal at which the object leaves the hand. This angle is represented by \(\theta\) in the figure shown. The distance, \(d,\) in feet, that the athlete throws is modeled by the formula $$ d=\frac{v_{0}^{2}}{16} \sin \theta \cos \theta $$ in which \(v_{0}\) is the initial speed of the object thrown, in feet per second, and \(\theta\) is the angle, in degrees, at which the object leaves the hand. a. Use an identity to express the formula so that it contains the sine function only. b. Use your formula from part (a) to find the angle, \(\theta,\) that produces the maximum distance, \(d,\) for a given initial speed, \(v_{0}\).

Step by step solution

01

Express With Sine Only

To express the given formula with only the sine function, we need to use the trigonometric identity: \( \sin2\theta = 2\sin\theta \cos\theta \). Rewriting the given equation, \( d = \frac{{v_0}^2}{16}\sin\theta\cos\theta \), we get \( d = \frac{{v_0}^2}{32}\sin 2\theta\).

02

Find Maximum Distance

To find the angle that maximizes the distance, we set the derivative of the equation to zero. The derivative of \( \sin 2\theta \) is \( \cos 2\theta \). Equating the derivative to zero gives us \( \cos 2\theta = 0 \), which implies \( 2\theta = 90^\circ \) or \( \theta = 45^\circ \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Trigonometric Functions in Throwing Events

Trigonometric functions are essential in modeling the physics behind throwing events in track and field. These functions help us understand how the angle and speed of an object affect its trajectory.

In the given exercise, the formula for distance is expressed in terms of sine and cosine functions:

• \( d = \frac{v_0^2}{16} \sin\theta \cos\theta \)

This formula uses both \( \sin \) and \( \cos \) to describe the angle \( \theta \) at which the object is thrown. By applying the identity \( \sin 2\theta = 2 \sin\theta \cos\theta \), the expression becomes simpler:

• \( d = \frac{v_0^2}{32} \sin 2\theta \)

This represents the distance using only the sine function, which is crucial for further calculations and understanding the core angle-related properties of the throw.

Calculating the Maximum Distance

Finding the angle that results in the maximum distance is a common problem in physics and mathematics. To achieve this, we need to consider the mathematical characteristics of the sine function.

The maximum value of \( \sin 2\theta \) is 1. Thus, to maximize the distance \( d \), the expression \( \sin 2\theta \) should equal 1.

• The derivative \( \cos 2\theta = 0 \) gives us clues on where this occurs.
• This condition leads to \( 2\theta = 90^\circ \), simplifying to \( \theta = 45^\circ \).
• At this angle, the distance is maximized for any given initial speed \( v_0 \).

Therefore, in the context of a throwing event, an angle of 45 degrees maximizes the distance, making it the optimal angle for throws when ignoring external factors like wind resistance.

Track and Field Mathematics Simplified

Mathematics in track and field events is not just about adding scores but also delving into physical concepts to optimize performance. In throwing events like javelin or shot put, understanding mathematical models can lead to improved results.

For athletes:

• Knowing that a 45-degree launch angle is optimal helps in training efforts by providing a target for technique refinement.
• Athletes and coaches can use this mathematical insight to experiment with actual throws, ensuring that training aligns with theoretical efficiency.

Field events are governed largely by straightforward mathematical rules, yet they offer a deep well of opportunity for strategizing and improving performance, making it a fascinating intersection of athletics and mathematics.