A boat heads S 15 degrees E on a river that flows due west. the boat travels S 11 degrees W with a speed of 25 km per hour. Find the speed of the current and the speed of the still water.
This is due tomorrow.
Draw a vector AB , S15°E, showing the boat in still water.
Draw BC, going West, showing the speed of the river.
Join AC , the resultant vector.
By simple calculation of angles,
angle A = 26°
angle B = 75°
angle C = 79°
I see two simple applications of the sine law
for the speed of the river:
a/sin26° = 25/sin75°
a = 25sin26/sin75 = 11.35
for the boat's speed in still water:
c/ain79 = 25/sin75
c = 25.406
To find the speed of the current and the speed of the still water, let's break down the given information step by step:
1. The boat heads S 15 degrees E on a river that flows due west.
- This means that the boat is moving in a direction that is 15 degrees east of south.
- The river is flowing directly west, so the current is pushing the boat in a direction perpendicular to the boat's heading.
2. The boat travels S 11 degrees W with a speed of 25 km per hour.
- This means that the boat is moving in a direction that is 11 degrees west of south.
- The speed of the boat with respect to the water is given as 25 km per hour.
To solve for the speed of the current and the speed of the still water, we need to use vector addition and trigonometry.
Step 1: Analyzing the boat's velocity:
- Convert the given directions into Cartesian coordinates. Let's assume the north direction is positive and the east direction is positive.
- To convert the boat's direction to Cartesian coordinates, we can use trigonometry:
- The boat's initial direction is 180° - 15° = 165° east of north.
- The boat's S 11 degrees W direction is 180° + 11° = 191° west of north.
- Using trigonometry, we can find the x and y-components of the boat's velocity:
- The x-component is given by Vx = V * cos(θ)
- The y-component is given by Vy = V * sin(θ), where V represents the speed of the boat with respect to the water.
- So, the boat's velocity V is equal to:
- Vx = V * cos(165°)
- Vy = V * sin(165°)
Step 2: Analyzing the current velocity:
- Since the river flows due west, its velocity is purely in the negative x-direction.
- Let's assume the speed of the current is C.
Step 3: Adding the boat's and current's velocities:
- The total velocity of the boat with respect to the ground is the vector sum of the boat's velocity and the current's velocity.
- The x-component of the total velocity is the sum of the x-components of the boat's and current's velocities.
- The y-component of the total velocity is the sum of the y-components of the boat's and current's velocities.
Step 4: Equating the components of the total velocity and solving for the unknowns:
- Equate the x-component of the total velocity to zero, since the boat doesn't move in the east-west direction:
- Vx + (-C) = 0
- Vx = C
- Equate the y-component of the total velocity to zero, since the boat doesn't move in the north-south direction:
- Vy = 0
Now, let's solve these equations to find the speed of the current (C) and the speed of the still water (V):
1. Equating the x-component of the total velocity to zero:
- V * cos(165°) + (-C) = 0
- V = C / cos(165°)
2. Equating the y-component of the total velocity to zero:
- V * sin(165°) = 0
From the second equation, we can deduce that either V = 0 or sin(165°) = 0. Since the boat is moving, V cannot be zero.
Therefore, sin(165°) = 0. Solving for sin(θ) = 0 will give us the value of θ:
- 165° = 180° - θ
- θ = 180° - 165°
- θ = 15°
Now, we can calculate the speed of the current and the speed of the still water using the value of θ obtained:
1. From V = C / cos(165°), plug in the value of θ:
- V = C / cos(15°)
2. Simplifying, we get:
- cos(15°) = 1 / √2 * (√6 + √2)
Therefore, the speed of the current (C) is V cos(15°), and the speed of the still water (V) is V / cos(15°).
You can now substitute the value of V for your given speed, and calculate the speed of the current and speed of the still water using the equations derived.