Within the jet stream, a plane flies at 220 m/s at 19 degrees south of west. The jet stream itself is moving at 130 m/s due north. Find the velocity of the jet stream relative to the ground.
To find the velocity of the jet stream relative to the ground, we need to calculate the vector sum of the plane's velocity and the jet stream's velocity.
First, let's break down the given velocities into their horizontal and vertical components.
The plane's velocity of 220 m/s can be broken down as follows:
- Horizontal component: 220 m/s * cos(19°) = 210.50 m/s
- Vertical component: 220 m/s * sin(19°) = 74.07 m/s (southward direction)
The jet stream's velocity of 130 m/s can be broken down as follows:
- Horizontal component: 0 m/s (since it's moving due north)
- Vertical component: 130 m/s
Now, let's find the horizontal and vertical components of the jet stream's velocity relative to the ground by adding the respective components of the plane and the jet stream.
Horizontal component: 210.50 m/s + 0 m/s = 210.50 m/s
Vertical component: 74.07 m/s - 130 m/s = -55.93 m/s (since it's moving southward against the jet stream)
Finally, to find the magnitude and direction of the resulting velocity, we can use the Pythagorean theorem and trigonometry.
Magnitude:
Velocity = √((210.50 m/s)^2 + (-55.93 m/s)^2) ≈ 219.53 m/s
Direction:
The direction can be found using the inverse tangent function:
Direction = arctan((-55.93 m/s) / (210.50 m/s)) ≈ -15.01°
Therefore, the velocity of the jet stream relative to the ground is approximately 219.53 m/s at an angle of about 15.01° south of west.