A boat travels across a wide river. if the boat can travel at a speed of 25 mph and its pilot heads in a direction of 30 degrees, while the current's velocity is 5.1 mph at a heading of 270 degrees, what will the final velocity of the boat be?
22mph
To find the final velocity of the boat, we need to break down the boat's velocity and the current's velocity into their horizontal and vertical components.
Given:
Boat's speed (Vb) = 25 mph
Boat's heading (θb) = 30 degrees
Current's speed (Vc) = 5.1 mph
Current's heading (θc) = 270 degrees
To calculate the horizontal component (Vx) and vertical component (Vy) for the boat's velocity, we can use trigonometry.
1. Calculate the horizontal component of the boat's velocity (Vbx):
Vbx = Vb * cos(θb)
= 25 * cos(30 degrees)
≈ 21.65 mph
2. Calculate the vertical component of the boat's velocity (Vby):
Vby = Vb * sin(θb)
= 25 * sin(30 degrees)
≈ 12.5 mph
3. Calculate the horizontal component of the current's velocity (Vcx):
Vcx = Vc * cos(θc)
= 5.1 * cos(270 degrees)
= 5.1 * 0
= 0 mph
4. Calculate the vertical component of the current's velocity (Vcy):
Vcy = Vc * sin(θc)
= 5.1 * sin(270 degrees)
= 5.1 * (-1)
= -5.1 mph
5. Calculate the final horizontal velocity (Vx_final):
Vx_final = Vbx + Vcx
= 21.65 + 0
= 21.65 mph
6. Calculate the final vertical velocity (Vy_final):
Vy_final = Vby + Vcy
= 12.5 + (-5.1)
= 7.4 mph
7. Calculate the magnitude of the final velocity (V_final):
V_final = sqrt(Vx_final^2 + Vy_final^2)
= sqrt((21.65)^2 + (7.4)^2)
~ 22.76 mph
Therefore, the final velocity of the boat will be approximately 22.76 mph.
To determine the final velocity of the boat, we need to consider both its speed and direction. The boat's velocity can be calculated by adding the velocity due to its own speed and the velocity due to the current.
First, we need to find the horizontal and vertical components of the boat's velocity. The horizontal component will determine the speed at which the boat moves across the river, while the vertical component will determine the speed at which the boat moves along the river.
Given:
Boat speed (Vb) = 25 mph
Boat direction (Θb) = 30 degrees
The horizontal component of the boat's velocity (Vbh) can be calculated using trigonometry:
Vbh = Vb * cos(Θb)
= 25 * cos(30 degrees)
≈ 21.65 mph
The vertical component of the boat's velocity (Vbv) is zero since it does not contribute to the boat's final velocity.
Next, let's consider the velocity of the current.
Current speed (Vc) = 5.1 mph
Current direction (Θc) = 270 degrees
The horizontal component of the current's velocity (Vch) is zero, as it is perpendicular to the direction in which the boat is heading.
The vertical component of the current's velocity (Vcv) can be calculated as:
Vcv = Vc * sin(Θc)
= 5.1 * sin(270 degrees)
≈ -5.1 mph
Note that the negative sign indicates that the current is flowing in the opposite direction to the boat's motion along the river.
Now, we can calculate the final velocity by adding the horizontal and vertical components of both the boat's velocity and the current's velocity:
Final horizontal velocity (Vfh) = Vbh + Vch
= 21.65 + 0
= 21.65 mph
Final vertical velocity (Vfv) = Vbv + Vcv
= 0 + (-5.1)
= -5.1 mph
The final velocity of the boat is the vector sum of its horizontal and vertical velocities:
Final velocity (Vf) = √(Vfh^2 + Vfv^2)
Vf ≈ √(21.65^2 + (-5.1)^2)
≈ √(469.0225 + 26.01)
≈ √495.0325
≈ 22.25 mph
Therefore, the final velocity of the boat will be approximately 22.25 mph, considering both its speed and the current velocity.