water flows in river 0.5km wide with speed 6km/hr. A boat can travel with speed 10km/hr in still water. In what direction should boat sail to reach opposite bank?
draw the figure. right triangle, hypotenuse 10, opposite side 6.
theta upstream=arcsin6/10
Vc + Vb = 6 + 10i = 11.7km/h[59o] CCW. = 31o E. of N.
Boat should sail 31o W. of N.
Vb - 6i = 10,
Vb = 10 + 6i = 11.7 km/h[31o] CCW = 31o N. of E.
To find the direction in which the boat should sail to reach the opposite bank, we need to consider the velocity of the river and the velocity of the boat.
Let's assume that the boat sails in a direction perpendicular to the river's flow. This means that the boat's velocity can be split into two components: the component parallel to the river's flow and the component perpendicular to the river's flow.
Given:
Width of the river = 0.5 km
Speed of the river's flow = 6 km/hr
Speed of the boat in still water = 10 km/hr
First, let's calculate the velocity of the river's flow relative to the boat. Since the boat is sailing perpendicular to the river's flow, the component of the river's velocity that affects the boat's motion is equal to the river's flow velocity itself.
Therefore, the velocity of the river's flow relative to the boat is 6 km/hr in the direction from one bank to the other.
Next, let's calculate the velocity of the boat relative to the ground. The boat's velocity can be split into two components: one parallel to the river's flow and one perpendicular to the river's flow.
The component of the boat's velocity parallel to the river's flow is the same as the river's flow velocity, which is 6 km/hr.
Now, to determine the overall velocity of the boat relative to the ground, we can use the Pythagorean theorem. The speed of the boat in still water is the hypotenuse, the river's flow velocity is one side, and the boat's velocity perpendicular to the river's flow is the other side of the right triangle.
Using the Pythagorean theorem, we can calculate the speed of the boat relative to the ground:
Velocity of the boat relative to the ground = sqrt((velocity of the boat in still water)^2 - (velocity of the river's flow)^2)
= sqrt((10^2) - (6^2))
= sqrt(100 - 36)
= sqrt(64)
= 8 km/hr
The overall velocity of the boat relative to the ground is 8 km/hr.
In order to reach the opposite bank, the boat needs to sail directly across the river. Therefore, the boat should sail perpendicular to the direction of the river's flow, with an angle of 90 degrees to the river's flow.
So, the boat should sail at a right angle to the river's flow (perpendicular to it) in order to reach the opposite bank.