/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 55 A submarine climbs at an angle o... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 55

A submarine climbs at an angle of \(30^{\circ}\) above the horizontal with a heading to the northeast. If its speed is 20 knots, find the components of the velocity in the east, north, and vertical directions.

Short Answer

Expert verified
Answer: The components of the velocity of the submarine in the east, north, and vertical directions are: East: \(5\sqrt{3}\) knots, North: \(5\sqrt{3}\) knots, and Vertical: 10 knots.

Step by step solution

01

Draw a diagram

Sketch the situation, indicating the east, north, and vertical directions, as well as the angle at which the submarine is climbing.
02

Split the velocity vector into horizontal and vertical components

First, we need to decompose the 20 knot velocity vector into its horizontal and vertical components. To do this, we can use the angle of \(30^{\circ}\) it makes above the horizontal plane. The horizontal component can be calculated by: \(v_h = v \cdot \cos{\theta}\) where \(v_h\) is the horizontal component of the velocity, \(v\) is the magnitude of the velocity (20 knots) and \(\theta\) is the angle (30 degrees). On the other hand, the vertical component \(v_v\) can be calculated as: \(v_v = v \cdot \sin{\theta}\)
03

Calculate the horizontal and vertical components

Now, plug in the given values to calculate the horizontal and vertical components: \(v_h = 20 \cdot \cos{30^{\circ}} = 20 \cdot \frac{\sqrt{3}}{2} = 10\sqrt{3}\) knots \(v_v = 20 \cdot \sin{30^{\circ}} = 20 \cdot \frac{1}{2} = 10\) knots
04

Split the horizontal component of the velocity into east and north components

Since the submarine is heading to the northeast, the horizontal component of the velocity will be equally split between the east and north directions. Hence, we can simply divide the horizontal component by 2 to get the components in the east and north directions. \(v_e = v_n = \frac{v_h}{2}\)
05

Calculate the east and north components

Now, plug in the value of the horizontal component to calculate the east and north components: \(v_e = v_n = \frac{10\sqrt{3}}{2} = 5\sqrt{3}\) knots
06

Write the final answer

The components of the velocity in the east, north, and vertical directions are as follows: East: \(5\sqrt{3}\) knots North: \(5\sqrt{3}\) knots Vertical: 10 knots

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