Velocity of a Boat A boat heads in the direction N 72^∘ E. The speed of the boat relative to the

step 1 So the problem tells us that a boat is moving at a 72 degree angle relative to the water at 24 m

Velocity of a Boat A boat heads in the direction N 72^∘ E....

Velocity of a Boat $A$ boat heads in the direction $\mathrm{N} 72^{\circ} \mathrm{E}$. The speed of the boat relative to the water is $24 \mathrm{mi} / \mathrm{h}$. The water is flowing directly south. It is observed that the true direction of the boat is directly east. (a) Express the velocity of the boat relative to the water as a vector in component form. (b) Find the speed of the water and the true speed of the boat.

Velocity of a Boat $A$ boat heads in the direction $\mathrm{N} 72^{\circ} \mathrm{E}$. The speed of the boat relative to the water is $24 \mathrm{mi} / \mathrm{h}$. The water is flowing directly south. It is observed that the true direction of the boat is directly east.
(a) Express the velocity of the boat relative to the water as a vector in component form.
(b) Find the speed of the water and the true speed of the boat.

Key Concepts

Trigonometric Functions in Vectors

Trigonometric functions, specifically sine and cosine, are used to convert a vector given in terms of magnitude and direction into its Cartesian components. This conversion is essential for working with vectors in a coordinate system and solving problems involving forces or velocities acting in multiple dimensions.

Vector Addition

Vector addition is the method used to combine two or more vectors to get a resultant vector. In problems involving velocities in different directions, the components of each vector are added together to determine the overall effect, such as the true velocity from the combination of the boat's motion and the current of the water.

Vector Decomposition

Vector decomposition is the process of breaking a vector down into its horizontal and vertical components using trigonometric functions such as sine and cosine. This allows for easier manipulation of vector quantities, as the behavior in each independent direction can be analyzed separately.

Relative Motion

Relative motion involves understanding the movement of an object as observed from different frames of reference. When multiple velocity vectors, such as the speed of a boat and the flow of water, combine, the resultant or true velocity is found using vector addition, reflecting how the object's movement is perceived relative to a stationary observer.

Transcript

Please give Ace some feedback