Problem 39
Question
A meteorologist tracks the movement of a thunderstorm with Doppler radar. At \(8: 00 \mathrm{PM},\) the storm was 60 mi northeast of her station. At 10: 00 PM, the storm is at 75 mi north. (a) The general direction of the thunderstorm's velocity is (1) south of east, (2) north of west, (3) north of east, (4) south of west. (b) What is the average velocity of the storm?
Step-by-Step Solution
Verified Answer
(a) 2 - north of west; (b) Average velocity is (-21.215, 16.285) mph.
1Step 1: Convert Positions into Coordinates
At 8:00 PM, the storm is 60 miles northeast of the station. For northeast, we assume equal north and east components. Therefore, the coordinates can be given as (42.43, 42.43) as both components equal \( \frac{60}{\sqrt{2}} \). At 10:00 PM, the storm is 75 miles north, giving us coordinates (0, 75).
2Step 2: Determine Direction
Convert the change in position to a vector. From (42.43, 42.43) to (0, 75), the vector becomes (-42.43, 32.57). The negative east component means the direction is **north of west**.
3Step 3: Calculate Change in Time
The time change from 8:00 PM to 10:00 PM is 2 hours.
4Step 4: Apply the Velocity Formula
The formula for average velocity is \( \vec{v} = \frac{\Delta \vec{r}}{\Delta t} \). The change in position \( \Delta \vec{r} \) is \((-42.43, 32.57)\). Divide by the time \(\Delta t = 2\).
5Step 5: Calculate Average Velocity
Average velocity is: \(\vec{v} = \left( \frac{-42.43}{2}, \frac{32.57}{2} \right) = (-21.215, 16.285)\) miles per hour.
Key Concepts
VectorsMeteorology PhysicsVelocity Calculation
Vectors
In mathematics and physics, vectors are essential tools used to describe quantities that have both magnitude and direction. This includes things like velocity, force, and displacement. Vectors are typically represented as arrows, where the length of the arrow corresponds to the vector's magnitude and the direction of the arrow points in the direction of the vector.
In two-dimensional space, vectors can be expressed in terms of their components along the x- and y-axes. For example, if a vector has components (x, y), it means the vector moves x units along the x-axis and y units along the y-axis. The magnitude of this vector can be calculated using the Pythagorean theorem as \( \sqrt{x^2 + y^2} \).
Vectors are added by combining their corresponding components. If you have two vectors, \( \vec{a} = (a_1, a_2) \) and \( \vec{b} = (b_1, b_2) \), their sum \( \vec{c} = \vec{a} + \vec{b} \) is given by \( \vec{c} = (a_1 + b_1, a_2 + b_2) \). Understanding how to break down vectors into their components and how to manipulate them is crucial in solving physics problems like the one presented in this exercise.
In two-dimensional space, vectors can be expressed in terms of their components along the x- and y-axes. For example, if a vector has components (x, y), it means the vector moves x units along the x-axis and y units along the y-axis. The magnitude of this vector can be calculated using the Pythagorean theorem as \( \sqrt{x^2 + y^2} \).
Vectors are added by combining their corresponding components. If you have two vectors, \( \vec{a} = (a_1, a_2) \) and \( \vec{b} = (b_1, b_2) \), their sum \( \vec{c} = \vec{a} + \vec{b} \) is given by \( \vec{c} = (a_1 + b_1, a_2 + b_2) \). Understanding how to break down vectors into their components and how to manipulate them is crucial in solving physics problems like the one presented in this exercise.
Meteorology Physics
Meteorology physics delves into the study of atmospheric phenomena. Key aspects include the movement of air masses and weather systems, such as thunderstorms, which are often tracked using radar technologies.
One of the tools frequently used in meteorology is Doppler radar. It helps meteorologists track storms by measuring the changes in the frequency of the returned radar signal, which is utilized to interpret the motion of the storm. When tracking a storm, as in the exercise, it is important to understand both where the storm started and its current position. This movement data helps predict the storm’s path.
Meteorologists use vector calculations to determine the direction and velocity of weather systems, which is vital for accurate weather forecasting. For example, while considering a thunderstorm moving from a northeast position to a more northern position, vector analysis helps determine its velocity and general movement direction.
One of the tools frequently used in meteorology is Doppler radar. It helps meteorologists track storms by measuring the changes in the frequency of the returned radar signal, which is utilized to interpret the motion of the storm. When tracking a storm, as in the exercise, it is important to understand both where the storm started and its current position. This movement data helps predict the storm’s path.
Meteorologists use vector calculations to determine the direction and velocity of weather systems, which is vital for accurate weather forecasting. For example, while considering a thunderstorm moving from a northeast position to a more northern position, vector analysis helps determine its velocity and general movement direction.
Velocity Calculation
Velocity is a vector quantity that refers to the rate at which an object changes its position. It's crucial for determining how fast something is moving and in which direction. The formula for average velocity is given by \( \vec{v} = \frac{\Delta \vec{r}}{\Delta t} \), where \( \Delta \vec{r} \) is the change in position vector and \( \Delta t \) is the interval of time over which this change occurs.
In the context of the meteorology exercise, calculating the velocity involves determining the change in the storm's position from 8:00 PM to 10:00 PM and dividing that by the time interval. The position changed from a northeast location represented by the vector coordinates (42.43, 42.43) to a northern position (0, 75).
This results in a position change or displacement vector of (-42.43, 32.57), indicating a movement mostly northward and slightly westward. Dividing these components by 2 hours provides the average velocity in the east-west and north-south directions. This understanding of velocity not only helps in physics challenges but is also practical in real-world applications like meteorological monitoring.
In the context of the meteorology exercise, calculating the velocity involves determining the change in the storm's position from 8:00 PM to 10:00 PM and dividing that by the time interval. The position changed from a northeast location represented by the vector coordinates (42.43, 42.43) to a northern position (0, 75).
This results in a position change or displacement vector of (-42.43, 32.57), indicating a movement mostly northward and slightly westward. Dividing these components by 2 hours provides the average velocity in the east-west and north-south directions. This understanding of velocity not only helps in physics challenges but is also practical in real-world applications like meteorological monitoring.
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