the speed of a powerboat in still water is 47 kph. the powerboat heads directly west across the messina river. the river flows northwest at 3.5 kph. find the resulting vector of the powerboat.
To find the resulting vector of the powerboat, we need to consider the vector sum of the boat's speed in still water and the river's speed.
Let's break down the motion into its components. The boat's speed in still water is 47 kph directly west, and the river's speed is 3.5 kph northwest. To simplify calculations, we can convert the northwest river velocity to its west and north components.
The northwest direction is 45 degrees between the northwest and west directions. Using trigonometry, we can find the north and west components of the river's velocity:
North component: 3.5 kph * sin(45°) ≈ 2.47 kph (considered positive as it is north)
West component: 3.5 kph * cos(45°) ≈ 2.47 kph (considered negative as it is opposite to the boat's west direction)
Now, we can add the boat's speed in still water (47 kph directly west) to the components of the river's velocity:
Resultant north component: 0 kph + 2.47 kph = 2.47 kph (north)
Resultant west component: -47 kph + (-2.47 kph) = -49.47 kph (west)
Therefore, the resulting vector of the powerboat is approximately 2.47 kph north and 49.47 kph west.
To find the resulting vector of the powerboat, we need to consider the speed and direction of both the powerboat and the river.
Let's break down the velocities into their respective components. The powerboat is moving directly west, so its velocity in the x-axis (horizontal direction) will be -47 kph. The river is flowing northwest, so its velocity can be split into two components: one in the x-axis (west-east direction) and one in the y-axis (north-south direction).
The x-component of the river's velocity will be -3.5 kph (since the river is flowing northwest, which is toward the west). But since we're considering motion in the east-west direction, we consider it to be positive. Therefore, the x-component of the river's velocity is +3.5 kph.
The y-component of the river's velocity is in the south direction, but since we only need to find the resulting vector of the powerboat (which is moving only in the x-axis), we can disregard this y-component.
Now, we can find the resulting velocity of the powerboat by adding the velocities of the powerboat and the river.
Resulting velocity in the x-axis = velocity of the powerboat in the x-axis + velocity of the river in the x-axis
Resulting velocity in the x-axis = -47 kph + 3.5 kph
Resulting velocity in the x-axis = -43.5 kph
Therefore, the resulting vector (velocity) of the powerboat is -43.5 kph in the westward direction.
draw the velocity vectors.
upstream hypotenuse is 47
downstream leg is 3.5
now figure the other leg's length using the Pythagorean Theorem