1. A ship has a top speed of 3.0 m/s in calm water. The current of the ocean tends to push the boat at 2.0 m/s on a bearing of due South. What will be the net velocity of the ship if the captain points his ship on a bearing of 55 North or West and applies full power?
To find the net velocity of the ship, we need to consider the vector addition of the ship's velocity and the current's velocity.
1. Start by determining the ship's velocity components based on the given bearing:
- If the captain points his ship on a bearing of 55 degrees North, the ship's velocity component in the North direction will be 3.0 * cos(55°).
- If the captain points his ship on a bearing of 55 degrees West, the ship's velocity component in the West direction will be 3.0 * sin(55°).
2. Next, add the ship's velocity components to the current's velocity component:
- The current's velocity component in the South direction is -2.0 (since it pushes the boat in the opposite direction).
- The current does not have a velocity component in the North or West direction, so their values will be 0.
3. Sum up the velocity components in each direction to get the net velocity of the ship:
- The North net velocity component = ship's North velocity component + current's North velocity component.
- The West net velocity component = ship's West velocity component + current's West velocity component.
- The South net velocity component = ship's South velocity component + current's South velocity component.
4. Finally, combine the net velocity components using vector addition to obtain the magnitude and direction of the net velocity vector.
To find the net velocity of the ship, we need to consider the individual velocities of the ship and the ocean current.
First, let's break down the given information:
- The top speed of the ship in calm water is 3.0 m/s (magnitude) and has no specific direction mentioned.
- The ocean current tends to push the boat at 2.0 m/s (magnitude) on a bearing of due South.
To determine the net velocity, we need to combine the velocities of the ship and the ocean current. This can be done using vector addition.
1. Start by representing the ship's velocity vector pointing north (since we are given a bearing of 55° North or West). Let's call it V_ship.
2. Now, we need to split the magnitudes of the ship's velocity into horizontal (east-west) and vertical (north-south) components by utilizing trigonometry.
Since it is a right triangle, we can use the sine and cosine functions to find the components:
- The vertical component (V_ship_north) is given by V_ship * sin(55°).
- The horizontal component (V_ship_west) is given by V_ship * cos(55°).
3. Next, we add the components of the ship's velocity to the ocean current's velocity to get the net velocity. Since the ship's velocity is northward and the current's velocity is southward, we need to subtract the southward component (2.0 m/s).
- The net vertical component (V_net_north) = V_ship_north - 2.0 m/s.
- The net horizontal component (V_net_west) = V_ship_west.
4. Finally, we combine the net vertical and horizontal components to obtain the net velocity vector (V_net) using the Pythagorean theorem.
- The magnitude of the net velocity (|V_net|) = sqrt((V_net_north)^2 + (V_net_west)^2).
- The direction of the net velocity can be found using tangent: arctan(V_net_north / V_net_west).
Calculating these values will give us the net velocity of the ship.
Vs = 3m/s[135o] + 2m/s[270o]
X = 3*cos135 = -2.12 m/s.
Y = 3*sin135 + 2*sin270o = 0.1213 m/s.
Tan Ar = Y/X = 0.1213/-2.12 = -0.05723
Ar = -3.275o = Reference angle.
A = -3.275 + 180 = 176.7o
Vs = X/cosA = -2.12/cos176.7=2.123 m/s.
[176.7o]