A plane travels 3 km due north, then 5 km due northeast. Represent these two displacements graphically and determine the resultant displacement by finding is magnitude and direction.
y component = North = 3 + 5 sin 45
x component = East = 5 cos 45
d = sqrt (x^2 + y^2)
tan T = y/x where T is angle above x axis
90-T = angle clockwise from north
To represent the two displacements graphically, we can use a vector diagram. Start by drawing a coordinate system where the north direction is upwards.
1. Draw an arrow pointing upwards to represent the first displacement of 3 km due north. Label this arrow as displcement A.
2. From the end of displacement A, draw another arrow pointing to the right and slightly upwards to represent the second displacement of 5 km due northeast. Label this arrow as displacement B, and make sure it is at a 45-degree angle to the x-axis.
Now, to find the resultant displacement, we can use vector addition. Here's how you can do it:
1. Break down both displacements into their x and y components.
- Displacement A:
x-component = 0 km (since it's purely north)
y-component = 3 km (since it's purely in the y-direction)
- Displacement B:
x-component = 5 km * cos(45°)
y-component = 5 km * sin(45°)
2. Add up the x and y components separately to get the resultant components.
- Resultant x-component = 0 km + 5 km * cos(45°)
- Resultant y-component = 3 km + 5 km * sin(45°)
3. Use these resultant components to find the magnitude and direction of the resultant displacement.
- Magnitude (R) = sqrt((Resultant x-component)^2 + (Resultant y-component)^2)
- Direction = tan^(-1)(Resultant y-component / Resultant x-component)
Now you can calculate the magnitude and direction of the resultant displacement.