A yacht is moving at 10kmh in a south easterly direction and encounters a 3kmh current from the north. Find the actual speed and direction of the yacht.
Using trig:
let the speed be x km/h
after making your diagram,
x^2 = 10^2+3^2 - 2(10)(3)cos 135°
= 151.426...
x = 12.3 km/h
angle:
using the sine law:
sinØ/3 =sin135/12.3055
sinØ = .17238...
Ø = 9.9266..
direction angle would be S (45-9.93)° E
direction would be S 35.07° E
If you know vectors
r = (10cos315, 10sin315) + (3cos270, 3sin270)
= (10√2/2, -10sin√2/2) + (0, -3)
= (7.07107, -7.07107) + (0,-3)
= (7.07107, -10.07107)
magnitude = √(7.07107^2+(-10.07107)^2)
= 12.3 , (same as above)
tanØ = -10.07107/7.07107 = -1.42426..
Ø = -54.93°
which would put in quad IV as 305.07° or S 35.07° E
To find the actual speed and direction of the yacht, we can apply the concept of vector addition.
Step 1: Resolve the velocity of the yacht and the current into their respective components.
The velocity of the yacht can be resolved into two components: one in the south direction and the other in the east direction. Since the yacht is moving in a south-easterly direction, we can use trigonometry to find the east and south components.
East component of yacht's velocity (Vx) = V × cos(θ)
South component of yacht's velocity (Vy) = V × sin(θ)
Given:
V (velocity of yacht) = 10 km/h
θ (angle of yacht's direction, relative to north) = 45°
Vx = 10 km/h × cos(45°)
Vx = 7.07 km/h (approximately)
Vy = 10 km/h × sin(45°)
Vy ≈ 7.07 km/h (approximately)
The velocity of the yacht can be represented as V = Vx i + Vy j, where i and j are unit vectors in the east and south directions, respectively.
The current has a velocity of 3 km/h in the north direction. Since the north direction is opposite to the south component, the current's velocity can be represented as Vc = 0 i - 3 j.
Step 2: Add the vectors representing the yacht's velocity and the current's velocity.
The actual velocity of the yacht can be represented as V_actual = V + Vc.
V_actual = (Vx i + Vy j) + (0 i - 3 j)
V_actual = (Vx + 0) i + (Vy - 3) j
V_actual = 7.07 i + (7.07 - 3) j
V_actual = 7.07 i + 4.07 j
The magnitude of the actual velocity (speed) of the yacht can be found using the Pythagorean theorem:
|V_actual| = sqrt((7.07)^2 + (4.07)^2)
|V_actual| ≈ 8.29 km/h (approximately)
The direction of the yacht's velocity can be found using the inverse tangent function:
Direction = tan^(-1)(Vy / Vx)
Direction = tan^(-1)(4.07 / 7.07)
Direction ≈ 29.54° (approximately)
Therefore, the actual speed of the yacht is approximately 8.29 km/h, and its direction is approximately 29.54° east of south.
To find the actual speed and direction of the yacht, we need to consider the velocity of the yacht due to its own motion and the velocity of the current.
Let's break down the velocities into their components:
Velocity of the yacht:
- Speed: 10 km/h
- Direction: South easterly
Velocity of the current:
- Speed: 3 km/h
- Direction: North
To find the actual velocity of the yacht, we need to add the vectors of its own velocity and the current velocity.
Step 1: Convert the velocities to their x and y components:
Velocity of the yacht:
- X component: speed * cos(angle)
- Y component: speed * sin(angle)
Velocity of the current:
- X component: speed * cos(angle)
- Y component: speed * sin(angle)
Step 2: Add the x and y components separately to obtain the resultant velocity:
Resultant x component = yacht x component + current x component
Resultant y component = yacht y component + current y component
Step 3: Calculate the magnitude and direction of the resultant velocity:
Magnitude = √(x^2 + y^2)
Direction = arctan(y/x)
Let's calculate the actual speed and direction of the yacht:
Velocity of the yacht:
- X component: 10 km/h * cos(45°) ≈ 7.071 km/h
- Y component: 10 km/h * sin(45°) ≈ 7.071 km/h
Velocity of the current:
- X component: 3 km/h * cos(180°) = -3 km/h (since the current is coming from the north)
- Y component: 3 km/h * sin(180°) = 0 km/h (since the current is coming from the north)
Resultant x component = 7.071 km/h + (-3 km/h) ≈ 4.071 km/h
Resultant y component = 7.071 km/h + 0 km/h ≈ 7.071 km/h
Magnitude = √(4.071^2 + 7.071^2) ≈ √(16.575 + 49.995) ≈ √66.570 ≈ 8.163 km/h
Direction = arctan(7.071/4.071) ≈ 59.036°
Therefore, the actual speed of the yacht is approximately 8.163 km/h, and its direction is approximately 59.036° (measured clockwise from the north).