A boat heading north crosses a wide river with a velocity 15 km/h relative to the water. The river has a uniform velocity of 8 km/h due east. Determine the boat's velocity with respect to the shore.
(Choose the nearest answer)
9 km/h, 50degree
17 km/h, 62degree
39 km/h, 40degree
v east = 8
v north = 15
|v| = sqrt(64 + 225) = 17
angle clockwise from north = cos^-1 15/17 = 31
angle counterclockwise from east (landlubber math way) = 90 -31 = 59
To determine the boat's velocity with respect to the shore, we can use vector addition.
Let's break down the boat's velocity into its horizontal and vertical components:
Horizontal component (relative to the water): 15 km/h (the boat's velocity)
Vertical component (relative to the water): 0 km/h (since the boat is moving purely in the horizontal direction, there is no vertical component)
Now, let's break down the river's velocity into its horizontal and vertical components:
Horizontal component (due east): 8 km/h (the river's velocity)
Vertical component (due north): 0 km/h (since the river is flowing purely in the horizontal direction, there is no vertical component)
To find the boat's velocity with respect to the shore, we need to add the boat's and the river's velocities (taking into account their respective directions):
Horizontal component (relative to the shore): 15 km/h + 8 km/h = 23 km/h
Vertical component (relative to the shore): 0 km/h + 0 km/h = 0 km/h
Using the Pythagorean theorem, we can find the magnitude (or speed) of the boat's velocity with respect to the shore:
Magnitude = sqrt((23 km/h)^2 + (0 km/h)^2) = sqrt(529 km^2/h^2) = 23 km/h (rounded to the nearest whole number)
The boat's velocity with respect to the shore is approximately 23 km/h. However, none of the given answer choices match this result.
To determine the boat's velocity with respect to the shore, we need to combine its velocity relative to the water with the velocity of the water itself. We can use vector addition to find the resultant velocity.
The boat's velocity relative to the water is given as 15 km/h north (assuming north is the positive y-direction). The velocity of the river is given as 8 km/h due east (positive x-direction).
To find the resultant velocity, we can use the Pythagorean theorem to find the magnitude and trigonometry to find the direction.
Magnitude (speed):
The magnitude of the resultant velocity is given by √(vx^2 + vy^2), where vx is the x-component (8 km/h) and vy is the y-component (15 km/h).
√(8^2 + 15^2) = √(64 + 225) = √289 = 17 km/h
Direction (angle):
The direction of the resultant velocity can be found using the inverse tangent function (tan^-1) with the formula θ = tan^-1(vy/vx), where θ is the angle formed by the vector with the positive x-direction.
θ = tan^-1(15/8) ≈ 62 degrees
Therefore, the boat's velocity with respect to the shore is approximately 17 km/h with a direction of 62 degrees.
Among the given options, the closest answer is: 17 km/h, 62 degrees.