A river flows due east at 1.42 m/s. A boat crosses the river from the south shore to the north shore by maintaining a constant velocity of 11.9 m/s due north relative to the water. If the river is 389 m wide, how far downstream is the boat when it reaches the north shore?
392
To find how far downstream the boat is when it reaches the north shore, we need to break down the motion of the boat into its horizontal and vertical components.
First, let's calculate the vertical component of the boat's velocity. The boat's velocity relative to the water is given as 11.9 m/s due north. Since there is no motion in the east-west direction, the horizontal velocity of the boat relative to the water is 0 m/s.
The river is flowing due east at 1.42 m/s. To determine the vertical component of the boat's velocity relative to the ground, we need to subtract the river's velocity from the boat's velocity.
Vertical component of boat's velocity relative to the ground = boat's velocity relative to the water - river's velocity
Vertical component = 11.9 m/s - 1.42 m/s = 10.48 m/s
Now, let's calculate the time it takes for the boat to cross the river. Since the boat's velocity relative to the ground is entirely vertical, we can use the distance and vertical speed to find the time.
Time = Distance / Vertical component
Time = 389 m / 10.48 m/s ≈ 37.13 seconds
Finally, we can calculate the distance the boat travels downstream during this time. Since the river's velocity is entirely horizontal, it does not affect the boat's east-west displacement.
Distance downstream = Time * River's velocity
Distance downstream = 37.13 s * 1.42 m/s ≈ 52.71 meters
Therefore, the boat is approximately 52.71 meters downstream when it reaches the north shore.