Problem 23 - Essential Calculus Early Transcendentals

Question

A quarterback throws a football with angle of elevation \(40^{\circ}\) and speed 60 \(\mathrm{ft} / \mathrm{s}\) . Find the horizontal and vertical components of the velocity vector.

Step-by-Step Solution

Answer

Horizontal: 45.96 ft/s, Vertical: 38.57 ft/s.

Step 1: Identify Given Values

The problem provides the angle of elevation \( \theta = 40^{\circ} \) and the speed \( v = 60 \text{ ft/s} \). These values will be used to determine the components of the velocity vector.

Step 2: Formula for Components

The horizontal component \( v_x \) of the velocity is calculated using \( v_x = v \cdot \cos(\theta) \). The vertical component \( v_y \) is found using \( v_y = v \cdot \sin(\theta) \).

Step 3: Calculate Horizontal Component

Substitute the given values into the horizontal velocity component formula: \( v_x = 60 \cdot \cos(40^{\circ}) \). Using a calculator, \( \cos(40^{\circ}) \approx 0.7660 \). Therefore, \( v_x \approx 60 \cdot 0.7660 = 45.96 \text{ ft/s} \).

Step 4: Calculate Vertical Component

Substitute the given values into the vertical velocity component formula: \( v_y = 60 \cdot \sin(40^{\circ}) \). Using a calculator, \( \sin(40^{\circ}) \approx 0.6428 \). Therefore, \( v_y \approx 60 \cdot 0.6428 = 38.57 \text{ ft/s} \).

Step 5: Conclusion

The horizontal component of the velocity is approximately \( 45.96 \text{ ft/s} \) and the vertical component is approximately \( 38.57 \text{ ft/s} \).

Key Concepts

Velocity Vector

A velocity vector helps us understand how an object moves. It is a way to express both the speed and direction of an object's motion. This vector is important in physics because it provides comprehensive details, not just about how fast something is moving, but also the direction in which it is moving. Velocity vectors have two main components:

• Magnitude: Represents the speed of the object, often measured in units like feet per second (ft/s) or meters per second (m/s).
• Direction: Given in degrees or radians, it tells us the path along which an object is moving, such as 40 degrees above the horizontal as seen in the example of a football pass.

By decomposing the velocity vector into its horizontal and vertical components, we gain detailed insights into the object's movement in both the horizontal and vertical planes, which is crucial for understanding and predicting its path.

Horizontal Component

The horizontal component of a velocity vector describes the object's motion along the horizontal plane. This component answers the question: "How far does the object travel horizontally at a given time?" To find this, we use the formula:\[ v_x = v \cdot \cos(\theta) \]Where:

• \(v_x\) is the horizontal component
• \(v\) is the total speed (magnitude of the velocity vector)
• \(\theta\) is the angle of elevation

In the quarterback's example, with a total speed of 60 ft/s and an angle of 40 degrees, the horizontal component would be 45.96 ft/s (using \(\cos(40^{\circ})\) which is approximately 0.7660). This tells us how effectively the velocity is used for moving the football along the ground.

Vertical Component

The vertical component of a velocity vector determines how the object moves vertically. Essentially, it answers: "How high does the object rise?" or "How much does it fall?" over time. This component primarily influences trajectory and height when the object is in motion. You can calculate it using the formula:\[ v_y = v \cdot \sin(\theta) \]Where:

• \(v_y\) is the vertical component
• \(v\) is the magnitude of the velocity vector
• \(\theta\) is the angle of elevation

For instance, in the case discussed, when a football is thrown at an angle of 40 degrees with speed 60 ft/s, the vertical component equals 38.57 ft/s (using \(\sin(40^{\circ})\) which is about 0.6428). This provides insight into how much the velocity contributes to the ball moving upward or downward.