Problem 68 - Precalculus

Question

Baseball. A baseball pitcher throws a ball with an initial velocity of 100 feet per second at an angle of \(5^{\circ}\) with the horizontal. What are the vertical and horizontal components of the velocity?

Step-by-Step Solution

Answer

Horizontal component: 99.62 ft/s, Vertical component: 8.72 ft/s.

Step 1: Understanding the Problem

We are given an initial velocity of a baseball, 100 feet per second, thrown at an angle of \(5^{\circ}\) to the horizontal. The task is to find the horizontal and vertical components of this velocity. These components can be found using trigonometric functions of the given angle.

Step 2: Horizontal Component Calculation

The horizontal component of velocity is calculated using the cosine function. We apply the formula: \( v_x = v \cdot \cos(\theta) \), where \( v = 100 \) feet per second, and \( \theta = 5^{\circ} \). Thus, \( v_x = 100 \cdot \cos(5^{\circ}) \).

Step 3: Solve for Horizontal Component

Calculate \( 100 \cdot \cos(5^{\circ}) \) using a calculator, which gives \( v_x \approx 100 \cdot 0.9962 = 99.62 \) feet per second. So, the horizontal component of the velocity is approximately 99.62 feet per second.

Step 4: Vertical Component Calculation

Similarly, the vertical component of velocity is calculated using the sine function. We use the formula: \( v_y = v \cdot \sin(\theta) \), where \( v = 100 \) feet per second, and \( \theta = 5^{\circ} \). Thus, \( v_y = 100 \cdot \sin(5^{\circ}) \).

Step 5: Solve for Vertical Component

Calculate \( 100 \cdot \sin(5^{\circ}) \) using a calculator, which gives \( v_y \approx 100 \cdot 0.0872 = 8.72 \) feet per second. Therefore, the vertical component of the velocity is approximately 8.72 feet per second.

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions are powerful mathematical tools used to relate angles and sides of triangles. In the context of physics, they help us break forces or velocities into components. We primarily use the sine (\(\sin\)) and cosine (\(\cos\)) functions to determine the vertical and horizontal components of a vector respectively.

For any vector at an angle \(\theta\) with respect to the horizontal, the horizontal component can be determined using the cosine function: \(v_x = v \cdot \cos(\theta)\), while the vertical component is found using the sine function: \(v_y = v \cdot \sin(\theta)\). These formulas assume that \(v\) is the magnitude of the vector.

Trigonometric functions are crucial for understanding how vectors resolve into components, which is common in many physical scenarios, such as projectile motion.

Initial Velocity in Motion

Initial velocity is the speed at which an object begins its movement, often crucial in determining its trajectory and eventual position. In the context of a thrown baseball, it acts as the starting speed and direction, telling us where the ball aims to travel initially.

Initial velocity is often split into two components:

• Horizontal Component (\(v_x\)): This dictates how far the object will travel horizontally.
• Vertical Component (\(v_y\)): This controls how high the object will go against gravity.

For a baseball thrown at a speed of 100 feet per second at an angle, its initial velocity determines how both components work together to affect the flight of the ball. Recognizing the initial velocity is key to predicting and understanding the ball's behavior throughout its motion.

Significance of Angle of Projection

The angle of projection is the angle at which an object is launched with respect to a reference line, typically the horizontal ground. It holds great importance in predicting the path an object, like a baseball, will take.

The angle affects how the initial velocity divides between horizontal and vertical components:

• Small angles result in a larger horizontal component, which means the object travels farther horizontally.
• Larger angles increase the vertical component, causing the object to rise higher but cover less horizontal distance.

For a pitcher throwing a baseball at a \(5^{\circ}\) angle, this relatively small angle favors a long horizontal reach with a modest upward trajectory. Understanding how the angle of projection influences the motion helps in strategizing throws to achieve desired distances and heights.