a man is driving his motor boat at 12 mph travelling N 30 E. The river currents are flowing due East at 8 mph. (A) express the true velocity of the motorboat as a vector. (B) what is the true speed of the motorboat? (C) What is the true direction of the motorboat?

Add the vectors

12N30E+8E
12sin30 N + 12 cos20E+8E
work that out into N, E vecots
xxxN + yyy E

true speed=sqrt (xxx^2 + yyy^2)

direction: arc tan (xxx/yyy) N of E

To answer these questions, we need to understand vector addition and the concept of relative velocity.

(A) To express the true velocity of the motorboat as a vector, we need to consider the contribution of both the boat's speed and the river's current.

Let's decompose the boat's velocity into two components: one in the North direction and the other in the East direction.

North component: The boat's speed is 12 mph in the N 30 E direction. To find the North component, we can use trigonometry. The N 30 E direction implies a 30-degree angle with the East direction. So, the North component is calculated as:

North component = Boat's speed * sin(angle) = 12 mph * sin(30°)

East component: The river's current flows due East at 8 mph. So, the East component of the boat's velocity is 8 mph.

Now, we can combine these components to find the true velocity of the boat as a vector:

True velocity = North component + East component

(B) To find the true speed of the motorboat, we need to calculate the magnitude of the true velocity vector.

True speed = magnitude of the true velocity vector

(C) To find the true direction of the motorboat, we need to calculate the angle that the true velocity vector makes with the East direction.

True direction = angle of the true velocity vector with the East direction

Now, let's calculate the values:

(A) True velocity of the motorboat as a vector:
North component = 12 mph * sin(30°)
East component = 8 mph
True velocity = North component + East component

(B) True speed of the motorboat:
True speed = magnitude of the true velocity vector

(C) True direction of the motorboat:
True direction = angle of the true velocity vector with the East direction