a boat heads directly across a river with a speed through the water 8m/s. the river current is 2m/s a. what is the magnitude and direction of the boat velocity relative to the earth?
b.when the boat reaches the other bank, how far will it be?
draw the velocity vectors. The resultant speed has magnitude v=√(2^2+8^2)
It drifts downstream by an angle θ such that tanθ = 2/8
without knowing the distance across the river, it's hard to say how far downstream it will have drifted. Also, you don't define exactly what you mean by "far."
To find the magnitude and direction of the boat velocity relative to the earth, we can use vector addition.
a. To find the magnitude of the boat's velocity relative to the earth, we need to add the boat's velocity through the water to the velocity of the river current. Given that the boat's speed through the water is 8 m/s and the river current is 2 m/s, we can use the Pythagorean theorem:
Magnitude = √((8 m/s)² + (2 m/s)²)
Magnitude = √(64 m²/s² + 4 m²/s²)
Magnitude = √(68 m²/s²)
Magnitude ≈ 8.25 m/s
b. To find how far the boat will be when it reaches the other bank, we can use the formula:
Distance = Boat's velocity relative to the earth * Time
Since the boat is heading directly across the river, the time taken to reach the other bank will be the same as the time taken to cross the river. Let's assume that the width of the river is given as 100 meters.
Time = Distance / Boat's velocity through the water
Time = 100 m / 8 m/s
Time ≈ 12.5 s
Distance = Boat's velocity relative to the earth * Time
Distance = 8.25 m/s * 12.5 s
Distance ≈ 103.12 m
Therefore, when the boat reaches the other bank, it will be approximately 103.12 meters away.
To find the magnitude and direction of the boat's velocity relative to the Earth, we need to consider the boat's velocity through the water (8 m/s) and the velocity of the river current (2 m/s).
a. Magnitude of Boat Velocity Relative to Earth:
To find the magnitude, we can use vector addition. The boat's velocity vector through the water and the river current vector form a right-angled triangle. By using the Pythagorean theorem, the magnitude of the boat's velocity relative to the Earth (vE) can be calculated as follows:
vE = √((vwater)^2 + (vcurrent)^2)
= √((8 m/s)^2 + (2 m/s)^2)
= √(64 m^2/s^2 + 4 m^2/s^2)
= √(68 m^2/s^2)
≈ 8.25 m/s
Therefore, the magnitude of the boat's velocity relative to the Earth is approximately 8.25 m/s.
b. Distance the Boat Travels:
To find the distance the boat will travel, we need to consider the time it takes for the boat to reach the other bank. Assuming the width of the river is given, we can calculate the time (t) using the formula:
t = d/vE
Where d is the width of the river.
For example, if the width of the river is 100 meters, and the magnitude of the boat's velocity relative to the Earth is 8.25 m/s (as calculated above), we can calculate the time as follows:
t = 100 m / 8.25 m/s
≈ 12.12 s
To find the distance the boat will travel, we need to multiply the time by the boat's velocity through the water:
distance = vwater * t
= 8 m/s * 12.12 s
≈ 96.96 m
Therefore, if the width of the river is 100 meters, the boat will travel approximately 96.96 meters when it reaches the other bank.