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Under certain conditions, tsunami war encountering land will develop into bores. A b is a surge of water much like what would expected if a dam failed suddenly and empti a reservoir into a river bed. In the case of a bc traveling from the ocean into a dry river b one study \({ }^{13}\) shows that the velocity \(V\) of the of the bore is proportional to the square root of height \(h\). Expressed in a formula, this is $$ V=k h^{0.5} \text {, } $$ where \(k\) is a constant. a. A bore travels up a dry river bed. How does the velocity of the tip compare with its initial velocity when its height is reduced to half of its initial height? b. How does the height of the bore compare with its initial height when the velocity of the tip is reduced to half its initial velocity? c. If the tip of one bore surging up a dry river bed is three times the height of another, how do their velocities compare?

Step by step solution

01

Interpret the formula for velocity

The velocity of the bore is given by the formula \( V = k h^{0.5} \), where \( k \) is a constant and \( h \) is the height of the bore. This means that the velocity of the bore is proportional to the square root of its height.

02

Understand part (a)

For part (a), we are asked how the velocity changes when the height is reduced to half of its initial value. Let the initial height be \( h_0 \), and initial velocity be \( V_0 = k h_0^{0.5} \). If the height is reduced to \( \frac{1}{2}h_0 \), the new velocity \( V' \) will be \( V' = k (\frac{1}{2}h_0)^{0.5} \).

03

Solve part (a) mathematically

Substitute \( \frac{1}{2}h_0 \) into the velocity equation: \[ V' = k ((\frac{1}{2})^{0.5} h_0^{0.5}) = k \frac{1}{\sqrt{2}} h_0^{0.5} = \frac{1}{\sqrt{2}} V_0 \]. Thus, the velocity becomes \( \frac{1}{\sqrt{2}} \) times the initial velocity \( V_0 \).

04

Understand part (b)

For part (b), we need to find how the height changes when the velocity is reduced to half of its initial value. If the new velocity is \( \frac{1}{2} V_0 \), substitute this into the velocity equation: \( \frac{1}{2} V_0 = k h'^{0.5} \).

05

Solve part (b) mathematically

Since \( V_0 = k h_0^{0.5} \), we know \( \frac{1}{2} V_0 = \frac{1}{2} k h_0^{0.5} = k h'^{0.5} \). Solving for \( h' \), we have: \[ h'^{0.5} = \frac{1}{2} h_0^{0.5} \].Squaring both sides, \( h' = \frac{1}{4} h_0 \), meaning the height is \( \frac{1}{4} \) its initial height.

06

Understand part (c)

In part (c), we are asked to compare the velocities of two bores where one is three times the height of the other. Let the height of the first bore be \( h \) and that of the second be \( 3h \). The velocities \( V_1 \) and \( V_2 \) are given by: \( V_1 = k h^{0.5} \) and \( V_2 = k (3h)^{0.5} \).

07

Solve part (c) mathematically

Substituting into the formula, \( V_2 = k (3)^{0.5} h^{0.5} = \sqrt{3} k h^{0.5} = \sqrt{3} V_1 \).Thus, the velocity of the bore with increased height is \( \sqrt{3} \) times the velocity of the other bore.

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Key Concepts

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

Tsunami

Tsunamis are massive waves caused by underwater disturbances, such as earthquakes, volcanic eruptions, or landslides. When these waves travel across the ocean, they may span long distances with persons on land being completely unaware. However, once they reach shallow coastal waters, these waves can surge dramatically, causing catastrophic consequences for coastal areas.
Tsunamis have distinct phases:

• Initiation: Triggered by seismic activity or underwater landslides.
• Propagation: The waves travel across the ocean at high speeds.
• Inundation: As they reach shallow waters or coastlines, they rise in height.

In the context of a riverbed, tsunami waves may form sudden surges known as bores, which move up river channels with great energy.

Bore Velocity

To understand the dynamics of bores, it is crucial to know how their velocity changes based on various factors. A bore is a surge, or a rapid increase in water level, that propagates through channels. The velocity of a bore, indicated as "V," is influenced by its height "h."

• Formula: The relationship is represented as \( V = k h^{0.5} \), where \( k \) is a constant.
• Proportional to height: The square root of the height impacts the velocity.

This means that if the height of a bore changes, its speed will change proportionally. For instance, a bore with decreased height will have a slower velocity, while an increase in the bore's height results in accelerated velocity. Thus, understanding and calculating bore velocity can aid in predicting the movement of water in river channels after significant wave events like tsunamis.

Proportional Relationships

The concept of proportionality is a fundamental element in algebra and mathematics at large. It indicates that two quantities change in relation to each other at a constant rate. With bore velocity, the principle of proportionality applies: the velocity is proportional to the square root of its height.

• Expression: \( V = k h^{0.5} \)
• "k" is the constant of proportionality, specific to each scenario.
• Implications: If height is decreased to half, velocity becomes \( \frac{1}{\sqrt{2}} \times \) initial velocity.

This proportional link allows us to predict changes. For example, when a bore's height triples, its velocity increases by the factor of \( \sqrt{3} \). Proportional relationships simplify complex scenarios into digestible parts, making them easier to analyze and comprehend.

Mathematical Modeling

Mathematical modeling is invaluable for illustrating real-world scenarios using mathematical structures. Modeling the velocity of bores during a tsunami involves formulating relationships between measurable quantities like velocity and height.

• Purpose: Predict behaviors and outcomes based on the model.
• Case study: Model the bore velocity as \( V = k h^{0.5} \) to explore changes in speed according to height variations.

These models serve as predictive tools that help us understand potential outcomes of natural phenomena, allowing for better preparation and response strategies. By examining different variables and constants in equations, mathematical models give insight into what might happen when conditions change, like bores running up riverbeds following tsunamis.