An object has three component velocities: 5mi/hr toward the east, 12mi/hr toward the south, and 8mi/hr toward the northwest. Find the resultant velocity of the object.
For resultant velocity, I got 6mi/hr
I don't understand how to find the direction angle and direction (ex. N of E)
See previous post: Fri, 10-10-14, 10:51 PM.
To find the resultant velocity, we need to combine the three component velocities. We can do this by using vector addition.
Step 1: Break down each velocity into its horizontal and vertical components.
- The velocity toward the east has a horizontal component of 5 mi/hr and a vertical component of 0 mi/hr.
- The velocity toward the south has a horizontal component of 0 mi/hr and a vertical component of -12 mi/hr (note the negative sign since it is directed south).
- The velocity toward the northwest can be broken down into horizontal and vertical components using trigonometry. The angle between the northwest direction and the horizontal axis is 45 degrees since northwest is halfway between north and west. We can use this angle to find the components as follows:
- Horizontal component = 8 mi/hr * cos(45°) = 8 mi/hr * 0.7071 ≈ 5.657 mi/hr
- Vertical component = 8 mi/hr * sin(45°) = 8 mi/hr * 0.7071 ≈ 5.657 mi/hr
Step 2: Add up the horizontal and vertical components separately to get the resultant horizontal and vertical components.
- Horizontal component = 5 mi/hr + 5.657 mi/hr = 10.657 mi/hr
- Vertical component = 0 mi/hr + (-12 mi/hr) + 5.657 mi/hr = -6.343 mi/hr
Step 3: Use the resultant horizontal and vertical components to find the magnitude and direction of the resultant velocity.
- Magnitude of resultant velocity = sqrt((10.657 mi/hr)^2 + (-6.343 mi/hr)^2) ≈ 12.4 mi/hr
- Direction angle = arctan(-6.343 mi/hr / 10.657 mi/hr) ≈ -31.3 degrees
The negative sign on the direction angle indicates that the resultant velocity is in the southwestern direction. To express this direction using the cardinal directions (e.g., N of E), we can say that the resultant velocity is about 31.3 degrees S of W.