A river flows at a speed of 4.60 m/s. A boat, capable of moving with a speed of 5.80 m/s in still water is rowed across the river at an angle of 53.0° to the river flow. Calculate the resultant velocity with which the boat moves and the angle that its resultant motion makes to the river flow.
Vr = 4.6 + 5.8[53o]=
4.6 + 5.8*Cos53 + 5.8*sin53 =
4.6 + 3.49 + 4.63i = 8.09 + 4.63i =
9.32m/s[29.8o].
Well, isn't this a "row" with the flow? Let's figure it out!
To find the resultant velocity, we need to consider both the velocity of the boat in still water and the velocity of the river flow. These two velocities combine to create the resultant velocity of the boat.
Using some trigonometry, we can break down the velocity of the boat into two components: one parallel to the river flow and one perpendicular to it. The parallel component is given by the equation:
Vp = Vboat * cosθ
where Vboat is the velocity of the boat in still water and θ is the angle it makes with the river flow. Plugging in the values, we get:
Vp = 5.80 m/s * cos(53.0°)
Calculating this, we find Vp to be approximately 3.67 m/s.
Now, let's find the perpendicular component. It is given by:
Vperp = Vboat * sinθ
Plugging in the values, we get:
Vperp = 5.80 m/s * sin(53.0°)
Calculating this, we find Vperp to be approximately 4.61 m/s.
Finally, we can find the resultant velocity using the Pythagorean theorem:
Resultant velocity = sqrt(Vp^2 + Vperp^2)
Plugging in the values, we get:
Resultant velocity = sqrt((3.67 m/s)^2 + (4.61 m/s)^2)
Calculating this, we find the resultant velocity to be approximately 5.92 m/s.
So, the resultant velocity of the boat is approximately 5.92 m/s. Now, let's find the angle the resultant motion of the boat makes with the river flow.
We can use the inverse tangent function to find this angle:
θresultant = arctan(Vperp / Vp)
Plugging in the values, we get:
θresultant = arctan(4.61 m/s / 3.67 m/s)
Calculating this, we find the angle to be approximately 52.9°.
So, the resultant velocity of the boat is approximately 5.92 m/s and the angle it makes with the river flow is approximately 52.9°.
Looks like the boat is "flowing" smoothly across the river with a little "twist" to the side!
To calculate the resultant velocity with which the boat moves, we need to use vector addition. The boat's velocity can be separated into two components: one parallel to the river flow and one perpendicular to the river flow.
Step 1: Calculate the component of the boat's velocity parallel to the river flow.
The component parallel to the river flow can be calculated using the formula:
V_parallel = V_boat * cos(θ)
Where V_boat is the boat's velocity in still water, and θ is the angle between the boat's motion and the river flow.
V_parallel = 5.80 m/s * cos(53°)
V_parallel = 5.80 m/s * 0.6018
V_parallel ≈ 3.4854 m/s
Step 2: Calculate the component of the boat's velocity perpendicular to the river flow.
The component perpendicular to the river flow can be calculated using the formula:
V_perpendicular = V_boat * sin(θ)
Where V_boat is the boat's velocity in still water, and θ is the angle between the boat's motion and the river flow.
V_perpendicular = 5.80 m/s * sin(53°)
V_perpendicular = 5.80 m/s * 0.7986
V_perpendicular ≈ 4.3603 m/s
Step 3: Use the Pythagorean theorem to find the resultant velocity.
The resultant velocity can be calculated using the formula:
V_resultant = √(V_parallel^2 + V_perpendicular^2)
V_resultant = √(3.4854^2 + 4.3603^2)
V_resultant = √(12.1332 + 19.0054)
V_resultant = √31.1386
V_resultant ≈ 5.574 m/s
Step 4: Calculate the angle that the resultant motion makes with the river flow.
The angle can be calculated using the formula:
θ_resultant = tan^(-1)(V_perpendicular / V_parallel)
θ_resultant = tan^(-1)(4.3603 m/s / 3.4854 m/s)
θ_resultant ≈ tan^(-1)(1.2519)
θ_resultant ≈ 51.25°
Therefore, the resultant velocity with which the boat moves is approximately 5.574 m/s, and the angle that its resultant motion makes to the river flow is approximately 51.25°.
To solve this problem, we can use vector addition. The velocity of the boat with respect to the ground is the vector sum of its velocity in still water and the velocity of the river flow.
First, let's break down the velocities into their horizontal and vertical components.
The velocity of the river flow is given as 4.60 m/s. Since there is no vertical component, the horizontal component of the river flow velocity is 4.60 m/s.
The velocity of the boat in still water is given as 5.80 m/s, and it is moving at an angle of 53.0° to the river flow. We can calculate the horizontal and vertical components of the boat's velocity.
Horizontal component of the boat's velocity = 5.80 m/s * cos(53.0°)
Vertical component of the boat's velocity = 5.80 m/s * sin(53.0°)
Now, let's add the horizontal components of the velocities:
Resultant horizontal velocity = Horizontal component of the river flow velocity + Horizontal component of the boat's velocity
= 4.60 m/s + (5.80 m/s * cos(53.0°))
Next, let's add the vertical components of the velocities:
Resultant vertical velocity = Vertical component of the boat's velocity
= 5.80 m/s * sin(53.0°)
Now, we can calculate the magnitude of the resultant velocity using the Pythagorean theorem:
Resultant velocity = sqrt((Resultant horizontal velocity)^2 + (Resultant vertical velocity)^2)
Finally, we can find the angle that the resultant vector makes with the river flow using the inverse tangent function:
Angle = arctan(Resultant vertical velocity / Resultant horizontal velocity)
Calculating these values:
Horizontal component of the boat's velocity = 5.80 m/s * cos(53.0°) ≈ 3.057 m/s
Vertical component of the boat's velocity = 5.80 m/s * sin(53.0°) ≈ 4.648 m/s
Resultant horizontal velocity = 4.60 m/s + 3.057 m/s ≈ 7.657 m/s
Resultant vertical velocity = 4.648 m/s
Resultant velocity = sqrt((7.657 m/s)^2 + (4.648 m/s)^2) ≈ 8.894 m/s
Angle = arctan(4.648 m/s / 7.657 m/s) ≈ 31.0°
Therefore, the resultant velocity of the boat is approximately 8.894 m/s, and the angle it makes with the river flow is approximately 31.0°.