A passenger on a boat moving at 1.70 m/s on a still lake walks up a flight of stairs at a speed of 0.60 m/s. The stairs are angled at 45° pointing in the direction of motion as shown in figure below. Write the vector velocity of the passenger relative to the water.
To find the vector velocity of the passenger relative to the water, we need to add the velocity vectors of the boat and the passenger.
First, let's find the x-components of the velocities:
Velocity of the boat in the x-direction = 1.70 m/s
Velocity of the passenger in the x-direction = 0.60 m/s
Since the stairs are angled at 45°, the y-components of the velocities will be the same as the x-components:
Velocity of the boat in the y-direction = 1.70 m/s
Velocity of the passenger in the y-direction = 0.60 m/s
Now, we can add the x and y-components of the velocities to get the vector velocity of the passenger relative to the water:
Vector velocity of the passenger relative to the water = (1.70 m/s + 0.60 m/s) i + (1.70 m/s + 0.60 m/s) j
Simplifying the equation, we get:
Vector velocity of the passenger relative to the water = 2.30 m/s i + 2.30 m/s j
Therefore, the vector velocity of the passenger relative to the water is 2.30 m/s in the x-direction and 2.30 m/s in the y-direction.
To find the vector velocity of the passenger relative to the water, we need to combine the velocities of the boat and the passenger.
Let's break down the velocities into their x and y components:
- The boat's velocity:
- vx = 1.70 m/s (horizontal component)
- vy = 0 m/s (vertical component)
- The passenger's velocity:
- vx = 0.60 m/s × cos(45°)
- vy = 0.60 m/s × sin(45°)
Now, let's add these components to find the relative velocity of the passenger with respect to the water.
- The x-component of the relative velocity:
- v_rx = vx_boat + vx_passenger = 1.70 m/s + 0.60 m/s × cos(45°)
- The y-component of the relative velocity:
- v_ry = vy_boat + vy_passenger = 0 m/s + 0.60 m/s × sin(45°)
Therefore, the vector velocity of the passenger relative to the water is:
v_r = v_rx î + v_ry ĵ