Hi!Question: A river flows due east at 1.70m/s.A boat crosses the river from the south shore to the north shore by maintaining a constant velocity of 8.0m/s due north relative to the water.
A.What is the velocity of the boat relative to shore?___m/s and ___degree(north of east)
B.If the river is 200m wide. how far downstream has the boat moved by the time reaches the north shore?____m
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To answer part A of the question, we can use vector addition to find the velocity of the boat relative to the shore. Since the boat's velocity is given in the north direction relative to the water and the river flows due east, we can break down the boat's velocity into its north (VN) and east (VE) components.
Given:
River velocity (VR) = 1.70 m/s due east
Boat velocity relative to water (VBW) = 8.0 m/s due north
To find the velocity of the boat relative to the shore, we need to add the velocities of the boat and river vectors. The resultant vector gives us the magnitude and direction of the boat's velocity relative to the shore.
Step 1: Find the east component of the boat's velocity.
VE = VR = 1.70 m/s
Step 2: Find the north component of the boat's velocity.
VN = VBW = 8.0 m/s
Step 3: Use vector addition to find the resultant velocity.
V = √(VE² + VN²)
V = √(1.70² + 8.0²) ≈ 8.25 m/s (rounded to two decimal places)
Step 4: Find the angle north of east.
θ = tan⁻¹(VN / VE)
θ = tan⁻¹(8.0 / 1.70) ≈ 77.4° (rounded to one decimal place)
Therefore, the velocity of the boat relative to the shore is approximately 8.25 m/s and it is about 77.4° north of east.
Moving on to part B of the question, we can determine how far downstream the boat has moved by the time it reaches the north shore. We can use the concept of relative velocity to solve this.
Given:
River velocity (VR) = 1.70 m/s due east
Time taken to cross the river (t) = Unknown
Distance of the river (d) = 200 m
Step 1: Find the time taken to cross the river.
Since the boat's velocity relative to the water is 8.0 m/s and the river is 200 m wide, the time taken to cross the river is:
t = d / VBW = 200 / 8.0 = 25 seconds
Step 2: Find the downstream component of the boat's velocity.
The downstream component is equal to the velocity of the river since the boat is crossing directly perpendicular to the river. Therefore, the downstream component of the velocity is:
Vdownstream = VR = 1.70 m/s
Step 3: Calculate the distance downstream using the equation:
Distance downstream = Vdownstream × t
Distance downstream = 1.70 × 25 = 42.5 meters
Therefore, the boat would have moved approximately 42.5 meters downstream by the time it reaches the north shore.