You are attempting to row across a stream in your rowboat. Your paddling speed relative to still water is 3.0 m/s (i.e., if you were to paddle in water without a current, you would move with a speed of 3.0 m/s ). You head off by rowing directly north, across the stream.
Incomplete.
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To determine your actual speed and direction, we need to consider the speed and direction of the current.
Let's assume the speed of the current is 1.5 m/s and it is flowing directly west.
To determine your actual speed and direction, we can use vector addition.
The speed of the boat relative to the ground can be found using the Pythagorean theorem:
Boat speed = √((Paddling speed)^2 + (Current speed)^2)
Boat speed = √((3.0 m/s)^2 + (1.5 m/s)^2)
Boat speed = √(9 m^2/s^2 + 2.25 m^2/s^2)
Boat speed = √11.25 m^2/s^2
Boat speed ≈ 3.35 m/s
The direction of the boat can be found using trigonometry:
Direction (θ) = arctan(Current speed / Paddling speed)
Direction (θ) = arctan(1.5 m/s / 3.0 m/s)
Direction (θ) = arctan(0.5)
Direction (θ) ≈ 26.57° (north of west)
Therefore, your actual speed is approximately 3.35 m/s, and you are rowing in a direction approximately 26.57° north of west.
To determine your overall speed and direction across the stream, we need to consider the speed and direction of the stream itself.
Let's assume the stream is flowing to the east with a speed of 2.0 m/s.
Since you are rowing directly north, and the stream is flowing to the east, these two motions can be considered perpendicular. We can use vector addition to find the resulting motion.
The magnitude of your overall velocity can be found using the Pythagorean theorem. The formula is:
overall velocity = square root of (velocity of rowboat * velocity of rowboat + velocity of stream * velocity of stream)
Plugging in the given values:
overall velocity = square root of (3.0 m/s * 3.0 m/s + 2.0 m/s * 2.0 m/s)
overall velocity = square root of (9 m^2/s^2 + 4 m^2/s^2)
overall velocity = square root of (13 m^2/s^2)
overall velocity ≈ 3.61 m/s (rounded to two decimal places)
The direction of your overall velocity can be found using trigonometry. Since you are rowing directly north and the stream is going to the east, we have a right triangle.
The angle between your paddle velocity and the north direction can be found using the inverse tangent function:
angle = inverse tangent(velocity of stream / velocity of rowboat)
angle = inverse tangent(2.0 m/s / 3.0 m/s)
angle ≈ 33.69 degrees (rounded to two decimal places)
Therefore, your overall velocity across the stream is approximately 3.61 m/s, at an angle of approximately 33.69 degrees north of directly east.