A truck moving north at a speed of 70 km/h. The exhaust pipe on top sends out a trail of smoke that makes a 20 degree angle east of south. If the wind is blowing toward the east. What is the wind speed at that location?
tan 20 =v/70
There should be a diagram to support it
I need the working
To determine the wind speed at that location, we can use vector addition. Let's break down the given information and calculate the wind speed:
1. The truck is moving north at a speed of 70 km/h.
2. The exhaust pipe emits smoke at an angle of 20 degrees east of south.
3. The wind is blowing toward the east (let's define this as the x-axis).
First, we need to find the vertical component of the truck's velocity, which is the north component. To do this, we use trigonometry:
Vertical Component = Trucker's Speed * sin(angle)
= 70 * sin(20 degrees)
≈ 23.87 km/h
The truck's vertical component of velocity is approximately 23.87 km/h.
Next, we need to find the horizontal component of velocity of the truck, which is the eastward component. Again, we use trigonometry:
Horizontal Component = Trucker's Speed * cos(angle)
= 70 * cos(20 degrees)
≈ 66.45 km/h
The truck's horizontal component of velocity is approximately 66.45 km/h.
Since the smoke trail is produced by the speed of the truck, we can assume that it moves at the same speed as the truck. Therefore, the smoke trail's velocity can be represented by the vertical and horizontal components calculated above.
Now, let's determine the wind speed. The wind speed vector can be represented by two components: the horizontal component (along the x-axis) and the vertical component (along the y-axis).
Since the wind is blowing towards the east (positive x-axis), and we know the truck is moving north (positive y-axis), we need to find the difference between the truck's horizontal component and the wind speed's horizontal component. This difference will give us the wind speed's horizontal component.
Wind's Horizontal Component = Truck's Horizontal Component - Horizontal Component of the Smoke Trail
= 0 - 66.45 km/h
= -66.45 km/h
Since we know the angle between the wind direction and the x-axis (east) is 90 degrees, we can use trigonometry to find the magnitude of the wind's horizontal component:
Wind's Horizontal Component = Wind Speed * cos(angle)
Wind's Horizontal Component = Wind Speed * cos(90 degrees)
–66.45 = Wind Speed * 0
Therefore, the horizontal component of the wind's speed equals 0.
Now we calculate the vertical component of the wind's speed. Since the wind is blowing toward the east, but there is no vertical component mentioned, we can assume it is 0:
Wind's Vertical Component = 0 km/h
Therefore, the wind speed at that location is 0 km/h.