A boat starting on one side of the river heads to the south with a speed of 1.5 m/s, the river flows to the east at .8 m/s. What is the resultant velocity to the side of the river? And if the river is 50 m wide calculate the displacement of the boat
Well, if the boat is heading south and the river is flowing east, I'm picturing a boat doing the Cha-Cha slide in the wrong direction. And it's definitely not getting any style points for that!
But let's get serious now. To find the resultant velocity, we can use Pythagoras' theorem. We have the boat's speed to the south (1.5 m/s) and the river's speed to the east (0.8 m/s). So, we square both of these speeds, add them together, and take the square root:
Resultant velocity = √((1.5)^2 + (0.8)^2)
Resultant velocity = √(2.25 + 0.64)
Resultant velocity = √2.89
Resultant velocity ≈ 1.70 m/s
Now, to calculate the displacement of the boat, we need to find out how long it takes for the boat to cross the river. Given that the river is 50 m wide and the boat's speed to the south is 1.5 m/s, we can divide the distance by the speed to find the time:
Time = Distance / Speed
Time = 50 m / 1.5 m/s
Time ≈ 33.33 s
The boat takes approximately 33.33 seconds to cross the river. Now, we find the displacement by multiplying the time by the boat's southward velocity:
Displacement = Time * Velocity
Displacement = 33.33 s * 1.5 m/s
Displacement ≈ 49.99 m
So, the displacement of the boat is approximately 49.99 meters. The boat might be zigzagging its way across the river, but at least it's making progress!
To find the resultant velocity of the boat with respect to the side of the river, we can use vector addition.
Let's assume that the direction towards the south is positive and the direction towards the east is negative.
The boat's velocity with respect to the ground is 1.5 m/s towards the south, and the river's velocity is 0.8 m/s towards the east.
To find the resultant velocity, we can add these two velocities as vectors:
Resultant velocity = boat's velocity + river's velocity
Using vector addition, we can determine the direction and magnitude of the resultant velocity.
The magnitude of the resultant velocity can be found using the Pythagorean theorem:
Resultant magnitude = √((1.5 m/s)^2 + (-0.8 m/s)^2)
Resultant magnitude = √(2.25 m^2/s^2 + 0.64 m^2/s^2)
Resultant magnitude = √(2.89 m^2/s^2)
Resultant magnitude = 1.7 m/s (approx)
To find the direction of the resultant velocity, we can use trigonometry. The angle (θ) can be found as:
θ = atan((river's velocity) / (boat's velocity))
θ = atan(-0.8 m/s / 1.5 m/s)
θ ≈ -29.74 degrees
Therefore, the resultant velocity to the side of the river is approximately 1.7 m/s at an angle of -29.74 degrees.
To calculate the displacement of the boat, we can use the formula:
Displacement = Resultant velocity × Time
Let's assume the boat crosses the river in a straight line. The time taken to cross the river can be calculated using the formula:
Time = Distance / River's velocity = 50 m / 0.8 m/s = 62.5 s
Displacement = 1.7 m/s × 62.5 s
Displacement ≈ 106.25 m
Therefore, the displacement of the boat is approximately 106.25 meters.
To find the resultant velocity of the boat relative to the side of the river, we can use vector addition.
Given:
- Boat's velocity relative to the ground (south direction): 1.5 m/s
- River's velocity relative to the ground (east direction): 0.8 m/s
We need to find the resultant velocity of the boat relative to the side of the river. The resultant velocity is the vector sum of the individual velocities.
We can break down the velocities into their components:
- Boat's velocity component in the east direction: 0 m/s (since it's moving purely south)
- Boat's velocity component in the south direction: 1.5 m/s
- River's velocity component in the east direction: 0.8 m/s
- River's velocity component in the south direction: 0 m/s (since it's flowing purely east)
Now, we can add the corresponding components together:
- Resultant velocity component in the east direction: 0 + 0.8 m/s = 0.8 m/s
- Resultant velocity component in the south direction: 1.5 + 0 m/s = 1.5 m/s
The resultant velocity of the boat relative to the side of the river is the vector (0.8 m/s east, 1.5 m/s south).
To calculate the displacement of the boat, we can use the formula: displacement = velocity x time.
Given:
- Width of the river: 50 m
The boat crosses the river perpendicularly, so the time it takes to cross the river is time = distance / velocity.
Time = 50 m / 1.5 m/s = 33.33 s
Now, we can calculate the displacement using the velocity component in the east direction, as the boat moves directly eastward during the time it takes to cross the river:
Displacement = Velocity east x Time = 0.8 m/s x 33.33 s = 26.67 m
Therefore, the displacement of the boat is 26.67 meters to the east side of the river.
velocity:
magnitude=sqrt(.8^2+1.5^2)
direction: arctan .8/1.5 from the normal to the East
Displacement=50/(.8/1.5) magnitude, same angle as above.