A straight river flows east at a speed of 5 km/h. A motorboat has a speed of 20 km/h relative to the river. A boater wants to arrive at a point on the north shore of the river directly opposite to the starting point. In what direction of northwest should the boat be headed?
Ans: sin−1 (1⁄4)
he wants his west component to be 5
20 cos x across to north
20 sin x west
20 sin x = 5
sin x = 5/20 = 1/4
x = 14.5 deg
To determine the direction in which the boat should be headed, we need to consider the velocities and the angle between them. Let's break down the problem step-by-step:
Step 1: Determine the velocity of the boat relative to the ground.
Since the river flows east at a speed of 5 km/h, and the boat has a speed of 20 km/h relative to the river, we can add these velocities to find the velocity of the boat relative to the ground.
Velocity of the boat relative to the ground = Velocity of the boat relative to the river + Velocity of the river
= 20 km/h east + 5 km/h east
= 25 km/h east
Step 2: Determine the angle between the boat's velocity and the north direction.
In this step, we need to find the angle between the east direction (the boat's velocity) and the north direction. We will use trigonometric functions to find this angle.
Angle = sin^(-1)(opposite/hypotenuse)
Opposite side = 5 km/h
Hypotenuse = 25 km/h (velocity of the boat relative to the ground)
Angle = sin^(-1)(5/25)
Angle = sin^(-1)(1/5)
Step 3: Convert the angle from radian to degrees.
The angle obtained in radians needs to be converted into degrees to express it in a more familiar form.
Angle in degrees = (Angle in radians) * (180/π)
Angle in degrees = sin^(-1)(1/5) * (180/π)
Angle in degrees ≈ 11.53 degrees
Step 4: Determine the direction in which the boat should be headed.
The direction in which the boat should be headed is northwest. This direction can be determined by subtracting the angle obtained in step 3 from 45 degrees (northeast direction).
Direction = 45 degrees - 11.53 degrees
Direction ≈ 33.47 degrees northwest
Therefore, the boat should be headed in the direction of northwest, at an angle of approximately 33.47 degrees.
To determine the direction in which the boat should be headed, we can break down the problem into two components: the speed of the river and the speed of the boat relative to the river.
First, let's consider the speed of the river. Since the river flows east at a speed of 5 km/h, we can represent its velocity as a vector of (5 km/h, 0 km/h). This means that the river's current is only in the eastward direction, and there is no northward or southward component to its flow.
Next, let's consider the speed of the boat relative to the river. The boat has a speed of 20 km/h relative to the river. We can represent this as a vector of (20 km/h, 0 km/h). Since the boat is moving directly opposite to the river's current, its velocity vector is entirely in the westward direction. Again, there is no northward or southward component to the boat's speed.
To determine the direction in which the boat should be headed, we need to find the vector sum of the river's velocity and the boat's velocity relative to the river. This will give us the resultant velocity vector, which will represent the direction and magnitude of the boat's movement across the river.
Since the only component of the boat's velocity vector is in the westward direction (-20 km/h), and the river's velocity vector has no northward or southward component, the resultant velocity vector will have a westward component and a northward component.
To find the magnitude of the northward component, we can use trigonometry. The magnitude of the northward component will be equal to the magnitude of the resultant velocity vector multiplied by the sine of the angle between the resultant velocity vector and the northward direction.
In this case, we want to find the angle between the resultant velocity vector and the northward direction, which will give us the direction in the northwest. Since the westward component of the boat's velocity vector and the eastward component of the river's velocity vector cancel each other out, the magnitude of the resultant velocity vector will be 0 km/h in the westward direction. Therefore, the angle we need to find is the angle between the northward direction and a vector with only a northward component.
Using the equation sin(theta) = opposite/hypotenuse, we can calculate the angle theta using the values of the opposite side (northward component) and the hypotenuse (resultant velocity magnitude). In this case, the northward component is 0 km/h and the resultant velocity magnitude is 20 km/h.
Therefore, sinθ = 0/20 = 0.
We know that sinθ = 1/4 (as stated in the answer). However, this is incorrect based on the problem statement. Since the northward component is 0 km/h, the angle between the resultant velocity vector and the northward direction is 0 degrees. Thus, the boat should be headed directly west, not northwest.