a man wishes to travel due north in order to move to the other side of the river which is 5km wide flowing due east at 3per hour . if he can row at 10 kilometers in still water .find either by scale or by calculation the direction in which he must held his boat in order to get to his destination directly opposite is starting point
Draw the velocity vectors. His upstream (west of north) angle θ is such that
sinθ = 3/10
To determine the direction in which the man must steer his boat to reach his destination directly opposite his starting point, we need to consider the combined effects of the river's current and the man's rowing speed.
First, let's analyze the situation and break it down step by step:
1. The man wishes to travel due north.
2. The river is flowing due east at a speed of 3 km/h.
3. The man can row at a speed of 10 km/h in still water.
4. The river is 5 km wide.
Now, let's calculate the speed and direction at which the man needs to row to reach the opposite bank.
Step 1: Find the speed of the river's current relative to the man's rowing speed.
Since the river is flowing east at 3 km/h, and the man can row at 10 km/h in still water, the effective speed of the river's current (relative to the man's boat) is 3 km/h to the west (opposite to the current).
Step 2: Use vector addition to find the resultant velocity.
To determine the resultant velocity, we need to add the vector of the river's current (westward) to the vector of the man's rowing speed (northward).
We can use the Pythagorean theorem to find the resultant speed:
Resultant speed = √(rowing speed^2 + current speed^2)
= √(10^2 + 3^2)
= √(100 + 9)
= √109 km/h
Step 3: Find the direction of the resultant velocity.
The angle that the resultant velocity vector makes with the north direction is given by the inverse tangent of the ratio of the northward component to the eastward component.
Direction angle (θ) = arctan(rowing speed / current speed)
= arctan(10/3)
≈ 73.74°
Therefore, the man needs to steer his boat at an angle of approximately 73.74° (measured clockwise from north) to reach his destination directly opposite his starting point.
Note: The calculation assumes that the river's current speed remains constant throughout the crossing.