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Chapter 1: Problem 69

The speed of an object is the magnitude of its related velocity vector. A football thrown by a quarterback has an initial speed of 70 mph and an angle of elevation of \(30^{\circ}\). Determine the velocity vector in mph and express it in component form. (Round to two decimal places.)

Short Answer

Expert verified
The velocity vector is (60.62, 35.00) mph in component form.

Step by step solution

01

Understand the Problem

We need to express the velocity of the football in component form, which involves breaking it down into horizontal and vertical components based on the initial speed and angle of elevation.
02

Set Up the Velocity Components

The velocity vector \(\mathbf{v}\) can be expressed as \((v_x, v_y)\), where \((v_x)\) is the horizontal component and \((v_y)\) is the vertical component of the velocity.
03

Calculate the Horizontal Component

The horizontal component of the velocity vector \((v_x)\) is calculated using the formula \((v_x = v \times \cos(\theta))\), where \((v)\) is the initial speed of the football and \(\theta\) is the angle of elevation. Given \(v = 70 \, \text{mph}\) and \(\theta = 30\degree\), we have: \[v_x = 70 \times \cos(30\degree).\] Use a calculator to find \(\cos(30\degree) = \sqrt{3}/2 \approx 0.866\), then \[v_x = 70 \times 0.866 \approx 60.62\text{ mph}.\]
04

Calculate the Vertical Component

The vertical component of the velocity vector \((v_y)\) is calculated using the formula \((v_y = v \times \sin(\theta))\). Using \(v = 70 \, \text{mph}\) and \(\theta = 30\degree\), we have: \[v_y = 70 \times \sin(30\degree).\] Use a calculator to find \(\sin(30\degree) = 1/2 = 0.5\), then \[v_y = 70 \times 0.5 = 35.00\text{ mph}.\]
05

Combine the Components

With \(v_x = 60.62 \, \text{mph}\) and \(v_y = 35.00 \, \text{mph}\), the velocity vector in component form is: \[\mathbf{v} = (60.62, 35.00)\text{ mph}.\] This represents the velocity of the football expressed as horizontal and vertical components.

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Key Concepts

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

Component Form
When discussing the movement of objects such as a football, representing its velocity using a velocity vector is crucial for understanding its motion. The component form of a velocity vector breaks down the vector into two distinct parts: the horizontal component and the vertical component.
The velocity vector can be denoted as \( \mathbf{v} = (v_x, v_y) \), with \( v_x \) representing the horizontal component and \( v_y \) the vertical component. By using component form, we can easily understand how fast the object is moving in both the horizontal and vertical directions.
This breakdown makes it clearer how much of the overall speed is directed horizontally and how much is directed vertically. This decomposition is particularly helpful in physics and engineering, where the separate effects of gravity and other forces need to be studied.
Horizontal Component
The horizontal component of a velocity vector represents how fast the object is moving along the horizontal plane. This component can be calculated using some basic trigonometry.
For an object thrown at an angle, the horizontal component \( v_x \) can be determined using the formula:
  • \( v_x = v \times \cos(\theta) \)
where \( v \) is the initial speed and \( \theta \) is the angle of elevation.
For example, if a football is thrown with an initial speed of 70 mph at an angle of 30掳, the cosine function tells us how to spread the speed along the horizontal direction. This computes as 70 mph multiplied by \( \cos(30掳) \), approximately 0.866. That gives us a horizontal velocity of roughly 60.62 mph.
Vertical Component
The vertical component of the velocity vector shows how much of the object's speed is directed upwards. It is crucial for calculating how high the object will travel in the air.
To find the vertical component \( v_y \), use this formula:
  • \( v_y = v \times \sin(\theta) \)
Here, \( v \) is the initial speed and \( \theta \) is the angle of elevation.
For our example football, thrown at 70 mph at an angle of 30掳, we use the sine function, which gives the proportion of speed directed vertically. The computation is 70 mph times \( \sin(30掳) \), which equals 0.5. Thus, the vertical velocity is calculated as 35 mph.
This component is vital for predicting how the projectile will react to gravity during its flight.

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