A boat leaves the shore of a stream. It starts directly across from point A, as
shown. The velocity of the boat is 5.2 m/s, at an angle θ = 25E as shown. The stream is 94
meters across, and is moving at 1.5 m/s. How far downstream from point A does the boat
land? Round your answer to the nearest meter.
Please help and show steps!
To find the distance downstream from point A where the boat lands, we need to consider the velocities of both the boat and the stream.
Step 1: Resolve the velocity of the boat into its horizontal and vertical components.
The horizontal component of the boat's velocity is given by: Vx = V * cos(θ)
The vertical component of the boat's velocity is given by: Vy = V * sin(θ)
where V is the velocity of the boat (5.2 m/s) and θ is the angle of the boat's velocity (25 degrees).
Vx = 5.2 * cos(25)
Vx ≈ 4.696 m/s
Vy = 5.2 * sin(25)
Vy ≈ 2.233 m/s
Step 2: Determine the time it takes for the boat to cross the river.
To find the time, we can use the horizontal component of the boat's velocity and the width of the stream.
Time = distance / velocity
Time = 94 m / 4.696 m/s
Time ≈ 20 seconds (rounded to the nearest second).
Step 3: Calculate the distance downstream traveled by the stream.
The distance downstream can be determined by multiplying the velocity of the stream by the time it takes for the boat to cross the river.
Distance Downstream = velocity of the stream * time
Distance Downstream = 1.5 m/s * 20 s
Distance Downstream = 30 meters.
Therefore, the boat lands approximately 30 meters downstream from point A.
To find the distance downstream from point A where the boat lands, we need to break down the velocities into their x and y components.
First, let's break down the velocity of the boat. The given velocity of the boat is 5.2 m/s at an angle of 25 degrees with respect to the stream.
The horizontal component of the boat's velocity (Vx) is given by Vx = V * cos(θ), where V is the magnitude of the velocity and θ is the angle.
Vx = 5.2 m/s * cos(25°)
Vx = 5.2 m/s * 0.9063
Vx ≈ 4.7199 m/s
The vertical component of the boat's velocity (Vy) is given by Vy = V * sin(θ), where V is the magnitude of the velocity and θ is the angle.
Vy = 5.2 m/s * sin(25°)
Vy = 5.2 m/s * 0.4226
Vy ≈ 2.197 m/s
Now, let's consider the velocity of the stream. The stream is moving at a constant velocity of 1.5 m/s in the horizontal direction.
Since the stream is only moving horizontally, the stream's velocity does not have any vertical component.
Now, we can find the total velocity of the boat relative to the shore. The total horizontal component of the velocity of the boat relative to the shore (Vr-x) is the sum of the boat's horizontal velocity component and the stream's horizontal velocity.
Vr-x = Vx + Vstream
Vr-x = 4.7199 m/s + 1.5 m/s
Vr-x ≈ 6.2199 m/s
Since the stream's velocity is only in the horizontal direction, the vertical components do not add up. Therefore, the total vertical component of the velocity of the boat relative to the shore (Vr-y) remains the same as the boat's vertical velocity component.
Vr-y = Vy
Vr-y ≈ 2.197 m/s
Now, we can use these components to find the time it takes for the boat to cross the stream. We can use the formula:
Time = Distance / Velocity
The distance the boat needs to cross is the width of the stream, which is given as 94 meters.
Time = 94 m / Vr-x
Time = 94 m / 6.2199 m/s
Time ≈ 15.0896 s
Now that we have the time it takes for the boat to cross the stream, we can find how far downstream the boat lands. The distance downstream (D) is given by:
D = Vr-y * Time
D = 2.197 m/s * 15.0896 s
D ≈ 33.1552 m
Rounding to the nearest meter, the boat lands approximately 33 meters downstream from point A.