Check out the motion of this person. How can we describe the motion? There are two elements to the motion: vertical, and horizontal.
Color code the motion
Let’s remove the visual details, and concentrate on the motions.
Blue is the path of the moving object.
Green shows the horizontal motion.
Red shows the vertical motion.
Here’s the take-away idea: The vertical, and horizontal motion are totally independent!
How can we describe such motion in physics?
Allow me to introduce Despicable Me’s Vector
Vocab: A component is one part of a larger whole.
Vectors are any quantity that have a direction and a magnitude (like force or momentum)
Vectors have X and Y components
Mathematically, the components act like shadows of the force vector on the coordinate axes.
Here we see a force vector on the (x, y) plane.
The force vector is white, the x-axis is red, the y-axis is green, the origin is white.
We force vectors with their tails at the origin.
The light is shining directly into the (x, y) plane.
We see no shadows from this view.
Below is the same scene from another viewpoint.
The light is now shining directly from above (shining straight down, parallel to the y-axis.)
Note the shadow of the vector on the x-axis:
it represents the x-component of the force vector.
Next we have the same situation except the direction of the light has changed.
The light now is shining from the right, parallel to the x-axis.
A shadow of the force vector can be seen on the y-axis.
This shadow, mathematically, is the y-component of the force vector.
We are back to a flat surface diagram below; it shows how these components can be drawn.
The black vector is the two dimensional force vector, labeled F.
The red vector is the x-component of the force vector, labeled Fx.
It is pronounced ‘F sub x’. Since ‘x’ is a subscript, it really looks like this:
The subscript’s position is often implied, as here:
The green vector is the y-component of the force vector, labeled Fy, pronounced ‘F sub y’.
The components of the force vector can also be arranged this way, forming a right triangle:
The sign of the components
The x-component of the force vector can be positive or negative.
If it points to the right, it is positive.
If it points to the left, it is negative.
The y-component of the force vector can be positive or negative.
If it points up, it is positive.
If it points down, it is negative.
When right triangle trigonometry is used, use your vector diagram to decide which way the components are pointing. Then assign the correct sign to your values, as a last step in your solution.
The right triangle trigonometry as presented here will always yield positive results. It is for finding the lengths of the legs of a right triangle, as one might do in geometry.
One important application of this principle is in the recreational sport of sail boating. A full lesson is presented here
also see Points of Sail
The horizontal (sideways) component of motion is independent of the vertical component of the motion. This isn’t intuitive so sometimes we’ll see a motion that doesn’t match our gut instincts. For instance, what is happening here?
Firing a projectile on a flat surface
Firing a projectile on a curved surface…like the Earth!
Vectors – Motion and Forces in Two Dimensions
Lesson 1 – Vectors: Fundamentals and Operations
Lesson 2 – Projectile Motion
Lesson 3 – Forces in Two Dimensions
Vectors and Projectiles
Drag a vector onto the canvas. Drag the arrowhead to change its direction. Repeat up to two more times and guess the direction of the resultant. Improve your skill at adding vectors using the head-to-tail method.
The Name That Vector Interactive is a skill building tool that presents users with 12 vector addition challenges. Twenty-five vectors are displayed on a grid; each challenge involves adding three of the vectors together to determine the resultant.
The Vector Guessing Game will challenge learner’s understanding of adding vectors.Two random vectors are displayed and learners must decide on the size and direction of the resultant.
The Vector Addition: Does Order Matter? Investigate with this app
The variable-rich environment of the Projectile Simulator Interactive allows a learner to explore a variety of questions associated with the trajectory of a projectile. Learners can modify the launch height, the launch angle, and the launch speed and observe the effect upon the trajectory.
Massachusetts Science Curriculum Framework 2016
1. Motion and Forces: Central Concept: Newton’s laws of motion and gravitation describe and predict the motion of most objects.
1.1 Compare and contrast vector quantities (e.g., displacement, velocity, acceleration force, linear momentum) and scalar quantities (e.g., distance, speed, energy, mass, work).
Common Core Standards for Mathematics – Grades 9-12
Number and Quantity – Vector and Matrix Quantities
- N-VM.1 Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments and use appropriate symbols for vectors and their magnitudes.
- N-VM.2 Find the components of a vector.