Kathir then drives an airplane is headed north with a constant velocity of 450 km/h. The plane encounters a wind from the west at 100 km/h. In what direction will the plane travel and in 3 h how far will the plane travel?
Draw a diagram. It should be clear that the resultant speed is
√(450^2 + 100^2) = _____
and distance = speed * time
The direction will be NE at an angle θ where
tanθ = 100/450
To determine the direction the plane will travel, we need to consider the effect of the wind on the plane's velocity. Since the wind is coming from the west, it will act against the plane's northward velocity.
Given that the plane has a constant velocity of 450 km/h north and the wind is blowing at 100 km/h from the west, we can treat these velocities as vectors. The northward velocity of the plane can be represented as 450 km/h in the positive y-direction (+450 km/h), and the westward velocity of the wind can be represented as 100 km/h in the negative x-direction (-100 km/h).
To find the resultant velocity vector, we can use vector addition. We add the x- and y-components of the velocities separately. Since the plane travels purely in the north and the wind pushes it exclusively in the west, there is no effect in the y-direction.
The resultant velocity can be determined using the Pythagorean theorem since the x- and y-components form a right triangle. The magnitude of the resultant velocity (V) can be found using the formula:
V = sqrt(Vx^2 + Vy^2)
In this case, since Vy (northward velocity) is zero, we only need to calculate the magnitude of Vx (westward velocity). Using the formula, we have:
Vx = -100 km/h
V = sqrt((-100 km/h)^2 + 0^2)
= sqrt(10000 km^2/h^2 + 0)
= sqrt(10000 km^2/h^2)
= 100 km/h
The magnitude of the resultant velocity is 100 km/h. Since Vx is negative, the resultant velocity points in the west direction. Hence, the plane will travel west.
To find how far the plane travels in 3 hours, we can simply multiply the magnitude of the resultant velocity by the time traveled:
Distance Travelled = Velocity x Time
Distance Travelled = 100 km/h x 3 h
Distance Travelled = 300 km
Therefore, in 3 hours, the plane will travel a distance of 300 km in the west direction.