a) To find the resultant velocity, we need to use vector addition. Since the motorboat is moving due west and the current is moving due south, we can consider their velocities as vectors in a coordinate plane.
Let's label the westward velocity of the motorboat as -10 m/s along the x-axis (negative because it is in the opposite direction of positive x-axis) and the southward velocity of the current as -6 m/s along the y-axis (again negative because it is in the opposite direction of positive y-axis).
To find the resultant velocity, we can use the Pythagorean theorem:
Resultant velocity = √((-10)^2 + (-6)^2) m/s
Resultant velocity = √(100 + 36) m/s
Resultant velocity = √136 m/s
So, the resultant velocity of the boat is approximately 11.66 m/s at an angle of approximately 32.1° south of west.
b) To find the time it takes the boat to cross the river, we can use the formula:
Time = Distance / Velocity
The distance to cross the river is given as 200 m and the velocity we calculated in part a) is approximately 11.66 m/s.
Time = 200 m / 11.66 m/s
Time ≈ 17.15 s
So, it takes approximately 17.15 seconds for the boat to cross the river.
c) To find how far downstream the boat is when it reaches the other side, we can use the formula:
Distance = Velocity × Time
The downstream velocity is the southward velocity of the current, which is -6 m/s. The time to cross the river we calculated in part b) is approximately 17.15 s.
Distance = -6 m/s × 17.15 s
Distance ≈ -102.9 m
Since the downstream distance is negative, it means the boat ends up 102.9 meters upstream on the opposite side.
To summarize:
a) The resultant velocity of the boat is approximately 11.66 m/s at an angle of approximately 32.1° south of west.
b) It takes approximately 17.15 seconds for the boat to cross the river.
c) The boat ends up approximately 102.9 meters upstream on the opposite side.