A sail boat is acted upon a water by a current and the wind. The velocity of the wind is 16km/h from the west, and the veolocity of the current is 12km/h from thr south, find the resultant of these 2 velocitie
Mathematicians drive me crazy when they make up navigation problems and get the conventions all wrong.
Wind directions generally describe the direction the air is from.
Current directions describe the direction the water flows toward.
If we do what I guess they want we can find the resultant wind velocity measured from the boat which is drifting at 12 km/hr north due to current
The boat therefore feels 12 km/hr north wind and 16 km/hr west wind
tan angle north of west = 12/16
angle north of west = 36.9 deg north of west
magnitude = sqrt (12^2+16^2) = 20 km/hr
To find the resultant velocity, we can use vector addition.
Step 1: Represent the wind velocity and current velocity as vectors.
Let's represent the wind velocity as Vw = 16 km/h towards the west, and the current velocity as Vc = 12 km/h towards the south.
Step 2: Draw vectors representing the wind velocity and current velocity.
You can draw a vector pointing westward with a length of 16 units to represent the wind velocity, and another vector pointing southward with a length of 12 units to represent the current velocity.
Step 3: Add the two vectors using vector addition.
To add the two vectors, place the tail of the second vector (current velocity) at the head of the first vector (wind velocity). Draw the resultant vector from the tail of the first vector to the head of the second vector.
Step 4: Measure the magnitude and direction of the resultant vector.
Using a ruler, measure the length of the resultant vector. Let's say the length is R km/h.
The direction of the resultant vector can be determined by measuring the angle it makes with a reference axis, such as the positive x-axis. Measure the angle clockwise from the positive x-axis. Let's say the angle is θ degrees.
Step 5: Write down the magnitude and direction of the resultant vector.
The magnitude of the resultant vector is R km/h, and its direction is θ degrees from the positive x-axis.
So, the resultant velocity of the sailboat is R km/h at an angle of θ degrees from the positive x-axis.
To find the resultant velocity of the sailboat, we need to consider the velocities of both the wind and the current. We can treat these velocities as vectors, which have both magnitude and direction.
Let's start by drawing a diagram to visualize the problem. We can define the east direction as positive x-axis and the north direction as positive y-axis. Given that the wind is coming from the west (negative x-axis) at a velocity of 16 km/h and the current is coming from the south (negative y-axis) at a velocity of 12 km/h, we can represent these velocities as follows:
Wind velocity vector (Vw) = -16 km/h * i (i indicates the x-axis direction)
Current velocity vector (Vc) = -12 km/h * j (j indicates the y-axis direction)
Now, to find the resultant velocity (Vr), we can add these two vectors together:
Vr = Vw + Vc
In other words, we need to add the x-components and the y-components separately. Let's calculate them:
x-component:
Vr_x = Vw_x + Vc_x = -16 km/h
(Note: There's no x-component in the current velocity vector since it is entirely in the y-axis direction.)
y-component:
Vr_y = Vw_y + Vc_y = -12 km/h
So, the resultant velocity vector is Vr = (-16 km/h) * i + (-12 km/h) * j.
To get the magnitude and direction of the resultant velocity, we can use the Pythagorean theorem and trigonometry. The magnitude (|Vr|) can be calculated as:
|Vr| = sqrt(Vr_x^2 + Vr_y^2)
= sqrt((-16 km/h)^2 + (-12 km/h)^2)
= sqrt(256 km^2/h^2 + 144 km^2/h^2)
= sqrt(400 km^2/h^2)
= 20 km/h
The direction of the resultant velocity (θ) can be calculated as:
θ = arctan(Vr_y / Vr_x)
= arctan((-12 km/h) / (-16 km/h))
= arctan(0.75)
≈ 36.87°
Therefore, the resultant velocity of the sailboat is approximately 20 km/h at an angle of 36.87° northwards from the east direction.