Round your answers to two decimal places. The net velocity...

Round your answers to two decimal places. The net velocity of a ship is the vector sum of the velocity imparted to the ship by its engine and the velocity of the wind. The engine propels the ship at a velocity of 20 miles per hour in the direction $S 35^{\circ} \mathrm{E}$. (a) What are the components of the velocity imparted to the ship by its engine? (b) If the wind is blowing from north to south at 12 miles per hour, find the magnitude and direction of the net velocity of the ship. (c) Rework part (b) for the case where the wind is blowing from north to south at 15 miles per hour.

Round your answers to two decimal places.
The net velocity of a ship is the vector sum of the velocity imparted to the ship by its engine and the velocity of the wind. The engine propels the ship at a velocity of 20 miles per hour in the direction $S 35^{\circ} \mathrm{E}$.
(a) What are the components of the velocity imparted to the ship by its engine?
(b) If the wind is blowing from north to south at 12 miles per hour, find the magnitude and direction of the net velocity of the ship.
(c) Rework part (b) for the case where the wind is blowing from north to south at 15 miles per hour.

Key Concepts

Resultant Vector Calculation

Once individual vectors are decomposed into their components and added together, the resultant vector's magnitude can be determined using the Pythagorean theorem, and its direction can be calculated using inverse trigonometric functions like the arctan function. This resultant vector represents the combined effect of multiple vectors, such as engine thrust and wind velocity, in both magnitude and direction.

Trigonometric Functions in Navigation

Trigonometric functions, especially sine and cosine, are used to relate angles to side lengths in a right triangle. In vector problems like this, the cosine of the given angle is used to find the component adjacent to the angle, and the sine is used to find the opposite component. These functions thus facilitate the conversion from a vector's magnitude and bearing to its component form.

Vector Decomposition

This concept involves breaking a vector with a given magnitude and a direction into its horizontal and vertical components. Using trigonometric functions, the cosine of the angle gives the component in one direction (typically along the x-axis) and the sine of the angle gives the component in the perpendicular direction (typically along the y-axis). This is essential in navigational problems where forces or velocities have both magnitude and direction.

Coordinate System

A standard Cartesian coordinate system is used where one axis (usually the x-axis) represents east-west directions and the other (usually the y-axis) represents north-south directions. This system helps to represent navigational vectors by aligning them with familiar horizontal and vertical directions, making it easier to apply vector operations.

Vector Addition

Vector addition is the process of combining two or more vectors to find their net effect. In the context of navigation, it involves adding the respective horizontal and vertical components of each vector. The sum of these components then represents the overall effect, or resultant vector, which gives both a new magnitude and a new direction.

Transcript

00:01 Okay, so this problem has several parts.

00:02 The first part just wants to know the vector components of the velocity imparted on a ship by its engine.

00:10 So we know that the direction is going to, well, let's do the magnitude.

00:15 So the engine is 22 miles per hour.

00:21 20 miles per hour.

00:26 So that's our magnitude of our vector.

00:28 And the direction of our vector is 35 degrees east of south.

00:32 So if we draw a quick diagram, 35 degrees east of south means that this angle is 35 degrees.

00:41 So if we add that to 270, we'll have that our angle that the ship is putting onto us, i guess, is 305 degrees.

00:54 So to calculate the components, we do u is equal to our magnitude times the cosine of our angle given and 20 times the sign of our angle given.

01:17 Okay, so we just have to put those in our calculator.

01:19 So 20 times cosine of 305 is 11 .47 miles per hour in the eastern direction.

01:37 And then for the sign, 16 .38 in the southern direction.

01:53 Okay, so there's part a.

01:57 Part b says if the wind is blowing from north to south at 12 miles per hour, so the wind, let's call it w, our vector for the, that will be none east to west and north to south means it's going to be negative 12 and then it says find the magnitude and direction of the net velocity of the ship so we have to add that to our u vector which we just got so our u vector was 11 .47 and negative 12 and negative 16 is negative 28 our decimal will stay the same at 0 .38 okay so the sum of our wind and engine w plus u is going to be 11 .47 i guess i already added the so it'll be negative 28 .38.

03:13 So that's the wind plus the engine so this part just needs to stay 16.

03:18 That was just me rewriting it's got ahead of myself so the engine would give us negative 16.

03:24 Okay so wind plus engine that's our vector components of that so we need to know the magnitude and direction of the net velocity so magnitude i would would be the sum of the squares.

03:39 And then take the square root.

03:53 And that's just calculator works.

03:55 And we go plug those in.

04:11 So the magnitude is going to be 30 .61 miles per hour.

04:24 And then our direction, we can just do tan inverse...

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