Problem 67 - Precalculus

Question

Baseball. A baseball player throws a ball with an initial velocity of 80 feet per second at an angle of \(40^{\circ}\) with the horizontal. What are the vertical and horizontal components of the velocity?

Step-by-Step Solution

Answer

The horizontal component of velocity is approximately 61.28 ft/s, and the vertical component is approximately 51.44 ft/s.

Step 1: Understanding the Components of Velocity

The velocity of the baseball can be broken down into two components: the horizontal component \(v_x\) and the vertical component \(v_y\). These components can be found using trigonometric functions.

Step 2: Find the Horizontal Component

The horizontal component \(v_x\) of the velocity is found using the cosine function:\[ v_x = v \cdot \cos(\theta) \]where \(v\) is the initial velocity and \(\theta\) is the angle. Plugging in the given values, we have:\[ v_x = 80 \cdot \cos(40^{\circ}) \]Calculate \(v_x\) to find the horizontal component.

Step 3: Find the Vertical Component

The vertical component \(v_y\) of the velocity is found using the sine function:\[ v_y = v \cdot \sin(\theta) \]Plugging in the known values, we get:\[ v_y = 80 \cdot \sin(40^{\circ}) \]Calculate \(v_y\) to find the vertical component.

Step 4: Calculate and Round Results

Using a calculator, we find:\[ v_x = 80 \cdot \cos(40^{\circ}) \approx 80 \cdot 0.766 = 61.28 \text{ feet per second} \]\[ v_y = 80 \cdot \sin(40^{\circ}) \approx 80 \cdot 0.643 = 51.44 \text{ feet per second} \]These are the horizontal and vertical components of the velocity respectively.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical tools that help us relate angles to side ratios in right-angled triangles. The two primary functions we deal with in breaking down vector components are sine and cosine.

• Cosine (\( \cos \theta \)): Used to find the horizontal component. It relates the adjacent side to the hypotenuse.
• Sine (\( \sin \theta \)): Used to find the vertical component. It relates the opposite side to the hypotenuse.

In our baseball example, the ball is thrown at an angle and speed that can be thought of as the hypotenuse in a triangle. Applying trigonometric functions here helps us break this velocity down into manageable parts.

Initial Velocity

The term 'initial velocity' refers to the speed and direction with which the baseball is thrown. It is represented as a vector, which means it has both a magnitude (how fast) and direction (at what angle).

• Magnitude: In this exercise, it's 80 feet per second.
• Direction: Given as 40 degrees from the horizontal.

Initial velocity is crucial because its components—horizontal and vertical—determine how the ball moves through the air. We use trigonometric functions to find these components from the initial velocity given.

Horizontal Component

The horizontal component of velocity, denoted as \(v_x\), is the part of the initial velocity acting along the horizontal direction. This component is crucial because it remains constant, assuming no air resistance, throughout the motion of the baseball.
To calculate it, we use the cosine function:\[v_x = v \cdot \cos(\theta)\]
Example Calculation: Given that the initial velocity \(v\) is 80 feet per second and the angle \(\theta\) is 40°,\[v_x = 80 \cdot \cos(40^\circ) \approx 61.28 \text{ feet per second}\]
This shows how the velocity of the ball splits into a component that keeps it moving forward.

Vertical Component

The vertical component of velocity, represented as \(v_y\), affects how high the baseball will go and how quickly it reaches that height before gravity pulls it down. This component is not constant as gravity acts on it.
Here, we use the sine function to find it:\[v_y = v \cdot \sin(\theta)\]
Example Calculation: With the initial velocity \(v\) set at 80 feet per second and the angle \(\theta\) at 40°,\[v_y = 80 \cdot \sin(40^\circ) \approx 51.44 \text{ feet per second}\]
This demonstrates how we can determine how much of the initial velocity propels the ball upwards.