A boat is travelling south at a rate of 18 km/h. A man walks across the deck from east to west at a rate of 16 km/h. Find the magnitude and direction of his velocity.
the magnitude is clearly √(16^2+18^2)
now just figure the angle. Draw a diagram.
To find the magnitude and direction of the man's velocity, we need to determine his resulting velocity vector by considering the velocity of the boat and the man's velocity on the deck.
Let's break down the problem into components:
1. The boat is travelling south at a rate of 18 km/h. This means the boat's velocity vector points directly towards the south and has a magnitude of 18 km/h.
2. The man walks across the deck from east to west at a rate of 16 km/h. This means the man's velocity vector points directly towards the west and has a magnitude of 16 km/h.
Now, we can find the resulting velocity vector of the man by adding the velocities of the boat and the man:
Resulting velocity vector = Boat's velocity vector + Man's velocity vector.
To add these vectors, we need to consider their directions and magnitudes separately.
The boat's velocity vector points south with a magnitude of 18 km/h, and the man's velocity vector points west with a magnitude of 16 km/h.
For magnitude: We can use the Pythagorean theorem to find the magnitude of the resulting velocity vector:
Resulting velocity magnitude = sqrt(Boat's velocity magnitude^2 + Man's velocity magnitude^2).
Plugging in the values, we get:
Resulting velocity magnitude = sqrt(18 km/h^2 + 16 km/h^2)
Therefore, the magnitude of the resulting velocity vector is approximately 24.166 km/h.
For direction: We can use trigonometry to find the direction of the resulting velocity vector.
tan(theta) = Opposite side / Adjacent side
tan(theta) = Man's velocity magnitude / Boat's velocity magnitude
theta = arctan(Man's velocity magnitude / Boat's velocity magnitude)
Plugging in the values, we get:
theta = arctan(16 km/h / 18 km/h)
Therefore, the direction of the resulting velocity vector is approximately 41.186 degrees west of south.
In conclusion, the magnitude of the man's velocity is approximately 24.166 km/h, and the direction is approximately 41.186 degrees west of south.