a man walks 10 meters in a direction of 20 degrees North of East, then 20 meters in a direction due west. What is his total displacement vector in component form
break this problem down to horizontal (x) and vertical (y) components! it'll probably help if you drew the problem out!
horizontal:
(10cos20) meters
20 meters
add the horizontal components together
vertical:
(10sin20) meters
perform Pythagorean theorem-
total displacement: sqrt[(horizontal)^2+(vertical)^2]
remember displacement is a vector so it also has direction! to find direction: tan^-1(vertical/horizontal)
opps! I forgot to put a negative in front of -20 meters under the horizontal component.
To find the total displacement vector in component form, we need to break down each movement into east-west (x-axis) and north-south (y-axis) components.
Let's start by analyzing the first movement of 10 meters in a direction of 20 degrees North of East.
1. Convert the angle to standard Cartesian coordinates:
- Since the angle is measured North of East, we need to subtract it from 90 degrees to get the angle with respect to the positive x-axis.
- Angle with respect to the positive x-axis = 90 degrees - 20 degrees = 70 degrees.
2. Determine the components of the 10-meter movement:
- The east-west (x-axis) component can be found using the cosine of the angle: x-component = 10 meters * cos(70 degrees).
- The north-south (y-axis) component can be found using the sine of the angle: y-component = 10 meters * sin(70 degrees).
Moving on to the second movement of 20 meters in a direction due west:
3. Determine the components of the 20-meter movement:
- Since the movement is due west, there is no north-south (y-axis) component, and the east-west (x-axis) component is simply -20 meters.
Now, we can calculate the total displacement vector in component form by adding the respective components:
Total east-west (x-axis) component = x-component of the first movement + x-component of the second movement.
Total north-south (y-axis) component = y-component of the first movement + y-component of the second movement.
Thus, the total displacement vector in component form is [Total east-west component, Total north-south component].
To find the total displacement vector in component form, we need to break down the given directions into their respective x and y components.
Let's start by breaking down the first part of the man's motion, where he walks 10 meters in a direction of 20 degrees North of East.
We can assume East is the positive x-direction and North is the positive y-direction. This means that the x-component of his motion will be determined by the cosine of the angle, and the y-component will be determined by the sine of the angle.
The x-component can be calculated as:
x1 = 10 * cos(20°)
The y-component can be calculated as:
y1 = 10 * sin(20°)
Now, let's break down the second part of the man's motion, where he walks 20 meters due west.
Since we are moving due west (in the negative x-direction), the x-component of this motion will be -20, and the y-component will be 0.
Now that we have the x and y components for both parts of his motion, we can calculate the total x and y components by simply summing them up.
Total x-component = x1 + (-20)
Total y-component = y1 + 0
Now we have the components of the total displacement vector. We can express this vector in component form as (Total x-component, Total y-component).