What is the instantaneous acceleration (both magnitude and direction) due to the coriolis force on an object moving at 160 m s^-1 to the west located at 17 degrees south, 86 degrees west?
To find the instantaneous acceleration due to the Coriolis force, we need to consider the object's velocity and its location on the Earth. The Coriolis force acts on moving objects due to the Earth's rotation.
First, we need to break down the object's velocity into its components in the Earth's reference frame. The given velocity is 160 m/s to the west. However, we must convert this velocity into the Earth's rotating frame.
From the given information, we know the object is located at 17 degrees south, 86 degrees west. Since the Earth is rotating from west to east, we need to add the Earth's rotational velocity to the object's velocity. The rotational velocity can be approximated as 0.4651 m/s for every degree of latitude.
Since the object is at 17 degrees south, the rotational velocity in the northward direction would be:
0.4651 m/s/degree * 17 degrees = 7.9007 m/s north.
Adding this component to the object's westward velocity gives the object's velocity in the Earth's rotating frame:
Westward component: 160 m/s to the west
Northward component: 7.9007 m/s to the north
Now that we have the object's velocity in the Earth's rotating frame, we can calculate the Coriolis acceleration. The formula for the Coriolis acceleration is:
ac = -2 * (v x ω)
where ac is the Coriolis acceleration, v is the velocity vector, and ω is the Earth's angular velocity vector.
Let's break down the calculation step by step:
1. Calculate the magnitude of the Coriolis acceleration:
|ac| = 2 * |v| * |ω| * sin(θ)
where |v| is the magnitude of the velocity vector, |ω| is the magnitude of the Earth's angular velocity vector, and θ is the angle between the velocity vector and the northward direction.
The magnitude of the Earth's angular velocity vector can be approximated as 0.0000727 rad/s.
2. Calculate the direction of the Coriolis acceleration:
The Coriolis acceleration is always perpendicular to both the velocity vector and the Earth's angular velocity vector. In this case, since the object is moving westward and northward, the Coriolis acceleration will be directed downward.
By calculating the magnitude and direction of the Coriolis acceleration using the formulas and values provided, we can determine the instantaneous acceleration due to the Coriolis force on the object.