a river flows due south with a speedof 2.0m/s.a man steers a motorboat across the river:his velocity relative to the river is 4.2m/s due east.the river is 800m wide.what is his velocity relative to the earth?
To find the man's velocity relative to the Earth, we can break down his motion into components and then add them together.
Let's label the given information:
- The river flows due south with a speed of 2.0 m/s.
- The man steers the motorboat with a velocity of 4.2 m/s due east.
- The river's width is 800 m.
We need to find the man's velocity relative to the Earth, which consists of two components: the component due to the river's flow and the component due to the man's steering.
1. Velocity due to the river's flow:
Since the river flows due south, the man's velocity relative to the river will have no effect on this component. Therefore, we can ignore the eastward velocity of 4.2 m/s for now.
The velocity due to the river's flow can be calculated using the formula:
Velocity = Distance / Time
The distance is the width of the river, which is 800 m. The time it takes for the man to cross the river is equal to the distance divided by the velocity of 2.0 m/s (since the man is moving perpendicular to the river's flow).
Time = Distance / Velocity
Time = 800 m / 2.0 m/s
Time = 400 s
Now we have the time it takes for the man to cross the river.
2. Velocity due to the man's steering:
This component is the eastward velocity of the man, which is 4.2 m/s. As mentioned earlier, this velocity is relative to the river.
Now, let's find the total eastward distance covered by the man:
Distance = Velocity x Time
Distance = 4.2 m/s x 400 s
Distance = 1680 m
3. Calculating the total velocity relative to the Earth:
To find the total velocity relative to the Earth, we need to add the two components together. The eastward component will be 1680 m, and the southward component will be the river's flow velocity, which is 2.0 m/s.
Using the Pythagorean theorem, we can calculate the resultant velocity:
Resultant Velocity = √(Eastward^2 + Southward^2)
Resultant Velocity = √(1680^2 + 2.0^2)
Calculating the resultant velocity:
Resultant Velocity = √(2822400 + 4)
Resultant Velocity ≈ √2822404
Resultant Velocity ≈ 1680.0003 m/s
Therefore, the man's velocity relative to the Earth is approximately 1680.0003 m/s in a direction that is a combination of east and slightly south.
V = 2m/s[270o] + 4.2m/s[0o]
X = 4.2 m/s.
Y = -2 m/s.
tanA = -2/4.2 = -0.47619.
A =-25.46o = 25.46o South of East.
V = x/cosA = 4.2/cos(-25.46) = 4.65 m/s.