A boat heads across an East-West river flowing at 3 m/s [W]. The boat starts at the South End and travels at 6 m/s [N25W].
What is the velocity (including direction) of the boat with respect to the ground?
Vbc = Vb + Vc = 6m/s[115o]
Vb - 3 = 6*cos115 + 6*sin115
Vb - 3 = -2.54 + 5.44i
Vb = 0.46 + 5.44i, Q1
Tan A = Y/X = 5.44/0.46 = 11.82609
A = 85.2o N. of E. = 4.8o W. of N. =
Direction.
Vb = Y/sin A = 5.44/sin 85.2 = 5.46 m/s.
To find the velocity of the boat with respect to the ground, we need to combine the velocity of the boat with respect to the river and the velocity of the river itself.
1. Let's first find the components of the boat's velocity:
- The boat travels at a speed of 6 m/s.
- The boat is heading in a direction N25W. This means it is 25 degrees west of north.
- To find the northward component of velocity, we can use the equation: Vn = V * sin(theta), where V is the speed of the boat and theta is the angle west of north.
- Vn = 6 m/s * sin(25 degrees) = 6 m/s * 0.4226 = 2.54 m/s.
- To find the westward component of velocity, we can use the equation: Vw = V * cos(theta), where V is the speed of the boat and theta is the angle west of north.
- Vw = 6 m/s * cos(25 degrees) = 6 m/s * 0.9063 = 5.44 m/s.
2. Now let's consider the velocity of the river flowing at 3 m/s westward.
- The river's velocity is constant and solely in the westward direction.
3. To find the velocity of the boat with respect to the ground, we need to add the velocities of the boat (with respect to the river) and the river:
- The northward component of the boat's velocity remains the same (2.54 m/s), as the river does not affect this component.
- The westward component of the boat's velocity gets added to the westward velocity of the river:
- 5.44 m/s (velocity of the boat) + (-3 m/s) (velocity of the river) = 2.44 m/s.
4. Now we have the northward component (2.54 m/s) and the westward component (2.44 m/s) of the boat's velocity with respect to the ground.
- To find the magnitude of the boat's velocity with respect to the ground, we can use the Pythagorean theorem:
- V = sqrt(Vn^2 + Vw^2) = sqrt((2.54 m/s)^2 + (2.44 m/s)^2) = sqrt(6.4516 + 5.9536) = sqrt(12.4052) = 3.52 m/s.
5. The direction of the boat's velocity with respect to the ground can be found using the inverse tangent function:
- Theta = tan^(-1)(Vn / Vw) = tan^(-1)(2.54 m/s / 2.44 m/s) = tan^(-1)(1.04) = 46.7 degrees.
Therefore, the velocity (including direction) of the boat with respect to the ground is 3.52 m/s [N46.7W].
To find the velocity of the boat with respect to the ground, we need to consider the velocities due to the river flow and the boat's own motion.
Let's break down the problem:
1. River flow velocity: The river is flowing at a speed of 3 m/s to the west. Since the boat is heading across the river, we need to subtract the river flow velocity from the boat's velocity to account for the effect of the river on the boat's motion. Therefore, the velocity of the river flow is (-3 m/s) in the West direction.
2. Boat's own velocity: The boat is moving at a speed of 6 m/s in a direction 25 degrees West of North (N25W). To determine the component of the boat's velocity in the North direction, we need to use trigonometry. The North component of the velocity (Vn) can be calculated by multiplying the boat's velocity by the cosine of the angle between the boat's direction and the North direction (25 degrees).
Vn = 6 m/s × cos(25°) ≈ 5.433 m/s
Since the boat is heading North and the river flow is in the West direction, we don't need to consider the East component of the boat's velocity for this problem.
3. Combining the velocities: To find the total velocity of the boat with respect to the ground, we need to add the boat's own velocity component in the North direction to the river flow velocity. Since the river flow velocity is towards the West, we will subtract it.
The overall velocity of the boat with respect to the ground is given by:
Velocity = Vn - River flow velocity
= 5.433 m/s - (-3 m/s)
= 8.433 m/s
Since the river flow is towards the West and the boat's velocity component is towards the North, the direction of the boat's velocity with respect to the ground will be North of West.