Wind is blowing toward south at 8m/s, a runner jogs west at 7 m/s. What is the velocity (speed and direction) of the air relative to the runner?
X = -7 m/s
Y = -8 m/s
tan Ar = -8/-7 = 1.14286
Ar = 48.8o = Reference angle.
A = 48.8 + 180 = 228.8o, CCW
V = -8/sin228.8 = 10.63 m/s[228.8o].
To find the velocity of the air relative to the runner, we need to combine the velocities of the wind and the runner using vector addition.
First, let's define the coordinate system. Let east be the positive x-direction and north be the positive y-direction.
Given:
Wind velocity = 8 m/s towards the south (negative y-direction)
Runner velocity = 7 m/s towards the west (negative x-direction)
To find the resultant velocity, we can add the x-components and y-components of the velocities separately.
1. X-Component:
The wind does not have any horizontal (x-component) velocity since it is blowing solely in the y-direction. Therefore, the x-component of the air velocity relative to the runner is 0 m/s.
2. Y-Component:
The wind has a vertical (y-component) velocity of 8 m/s towards the south (negative y-direction), and the runner has no vertical velocity. Therefore, the y-component of the air velocity relative to the runner is -8 m/s.
Therefore, the velocity of the air relative to the runner is:
Speed: √(0^2 + (-8)^2) = 8 m/s
Direction: It is blowing towards the south (negative y-direction) relative to the runner.
To find the velocity of the air relative to the runner, we need to consider their directions and speeds.
Given:
- Wind velocity = 8 m/s towards the south
- Runner velocity = 7 m/s towards the west
To determine the relative velocity vector, we can use vector addition. We add the runner's velocity vector to the negative of the wind's velocity vector because we want to find the difference between the two velocities.
Let's break down the velocities into their components:
- Wind velocity: 8 m/s towards the south can be expressed as (0 m/s, -8 m/s) in terms of its x and y components.
- Runner velocity: 7 m/s towards the west can be expressed as (-7 m/s, 0 m/s) in terms of its x and y components.
Now, we add the respective components:
Velocity of the air relative to the runner = (wind velocity) + (-runner velocity)
Adding the x-components:
= (0 m/s) + (-7 m/s) = -7 m/s
Adding the y-components:
= (-8 m/s) + (0 m/s) = -8 m/s
Therefore, the velocity of the air relative to the runner is (-7 m/s, -8 m/s). This means the air is moving at a speed of 7 m/s to the west and 8 m/s towards the south with respect to the runner.