Solving for The Relative Motion of a Boat: Angle & Speed

[omerr]

Homework Statement

suppose the boat is initially at x = 0, y = 0. and the dock is located at x = 55 m, y = 11.7 m relative to this starting point. Assume that the speed of the boat relative to the water is 5.0 m/s and that the speed of the water relative to the ground is 1.4 m/s.
(a) At what angle relative to the x-axis must the boat be pointed in order to reach the other dock?
(b) With the angle found in part (a), what is the speed of the boat relative to the ground?

Homework Equations

The Attempt at a Solution

 

[tiny-tim]

Welcome to PF!

Hi omerr! Welcome to PF!

Hint: velocity is a vector, so velocities obey the law of vector addition.

So draw a vector triangle (and tell us what you get).

 

[thomate1]

To solve for the relative motion of the boat, we must first understand the velocity vectors involved. The boat has a velocity vector that is a combination of two velocities: its own velocity relative to the water and the water's velocity relative to the ground. This can be represented by the equation:

Vboat = Vboat relative to water + Vwater relative to ground

Using the given values, we can write this as:

Vboat = 5.0 m/s + 1.4 m/s = 6.4 m/s

(a) To determine the angle at which the boat must be pointed in order to reach the other dock, we can use trigonometry. We know that the x-component of the boat's velocity will be 6.4 m/s (since the boat is initially at x = 0 and the dock is at x = 55 m), and the y-component of the velocity will be 11.7 m/s (since the dock is at y = 11.7 m relative to the starting point of the boat).

Using the formula tanθ = opposite/adjacent, we can solve for the angle:

tanθ = 11.7 m/s / 6.4 m/s

θ = tan^-1 (11.7/6.4) = 59.3°

Therefore, the boat must be pointed at an angle of 59.3° relative to the x-axis in order to reach the other dock.

(b) To find the speed of the boat relative to the ground, we can use the Pythagorean theorem. The x-component of the boat's velocity is 6.4 m/s and the y-component is 11.7 m/s, so the magnitude of the boat's velocity can be found as:

Vboat = √(6.4^2 + 11.7^2) = 13.3 m/s

Therefore, the speed of the boat relative to the ground is 13.3 m/s.