A river flows due east at 1.34 m/s. A boat crosses the river from the south shore to the north shore by maintaining a constant velocity of 11.3 m/s due north relative to the water.
To determine the boat's actual velocity and direction relative to the ground, you need to use vector addition.
Step 1: Draw a diagram. Draw a river flowing eastward horizontally. Label the boat's velocity relative to the water as 11.3 m/s straight north. Draw an arrow representing the river's velocity going eastward.
Step 2: Add the vectors. Since vectors are added tip-to-tail, place the tail of the river's velocity vector at the tip of the boat's velocity vector. The resultant vector, connecting the initial and final points, gives you the boat's actual velocity relative to the ground.
Step 3: Measure the magnitude (speed) and direction of the resultant vector. Use the Pythagorean theorem to find the magnitude of the resultant vector:
Resultant speed = √(eastward velocity^2 + northward velocity^2)
Resultant speed = √(1.34^2 + 11.3^2) = √(1.7956 + 127.69) = √129.4856 = 11.38 m/s
To find the direction, use trigonometry. The direction of the resultant vector (angle θ) relative to the east direction is given by:
θ = arctan(northward velocity / eastward velocity)
θ = arctan(11.3 / 1.34) = arctan(8.4328) = 81.7°
Note: The direction is measured counterclockwise from the east direction.
So, the boat's actual velocity relative to the ground is approximately 11.38 m/s at an angle of 81.7° north of east.