Motion and Vectors: Kinematics Fundamentals
(Level 1)
Before you can analyze why something moves, you need to describe how it moves. Position. Velocity. Acceleration. Direction. These aren't just textbook terms—they're the vocabulary you use to communicate about any mechanical system that isn't sitting still. Get this wrong and everything downstream (forces, energy, dynamics) becomes guesswork.
What You'll Learn
Scalars and Vectors in Engineering Motion
A scalar is a number; 5 meters, 10 seconds, 50 kg are all scalars. A scalar has a magnitude but no direction or size. A vector has magnitude and direction. An example of a vector is 5 meters east, 10 m/s upward, 50 N to the right. There are many different types of vectors including forces, velocities, and accelerations. As engineers, we generally treat these physical quantities as if they occurred solely by magnitude, though, of course, in reality they always have a specific direction as well.
Velocity is a vector quantity with components of 3 m/s in the x (east) direction, and -4 m/s in the y (north) direction.
A vector tells you in what direction something is going, but a scalar does not.
MESS UP THE DIRECTION OF ANY QUANTITY FOR WHICH THE DIRECTION IS PHYSICALLY MEANINGFUL AND YOU CAN STILL ENSURE THAT YOUR CALCULATIONS ARE Mathematically correct, but PHYSICALLY MEANINGLESS. All quantities involving forces, momentum, torque, and fluid flow are vectors. And if you can't think in vectors, this book will be a headache for you.
Position, Displacement, and Distance in Engineering Motion
Position: where you are (relative to some origin).
Displacement: how far you've traveled (a vector, includes direction).
Distance: how far you have actually traveled (a scalar, includes no direction, only the total length of travel).
• When crawling: 2 meters out + 2 meters back + 2 meter crawl across = 6 meters.
• You ended up going no where.
Why This Matters Later - Displacement
Displacement determines how systems change state. You can have zero displacement while traveling 100 miles, but that is fundamentally different from traveling 100 miles in a straight line down a road. The distance traveled does not give information about the direction of that travel, so therefore it does not give information about how the position changed.
Velocity and Speed in Vector Motion
Velocity is the change of position with time. It is a vector quantity because it specifies not only the speed of an object, but also its direction of travel. Note that speed is different from velocity: speed is the magnitude of the velocity vector and thus is a measure of how fast one is traveling, without regard to the direction in which the movement is taking place.
You are currently moving at 4 m/s in the x direction and 3 m/s in the y direction.
Speed = √(4² + 3²) = √25 = 5 m/s.
Velocity has both a magnitude (Speed) and a direction. Speed is always the length of the velocity vector. Therefore, a velocity of 5 m/s east is equivalent to a velocity of 5 m/s in the direction you want to go.
This is a distinction that comes up a lot. When you are working with forces, you are working with changes in velocity, including changes in both speed and direction. This means that something could be traveling at a constant speed and still have a force acting on it, if that force is causing the velocity to change in some way—indeed, the velocity could be changing purely in direction, as it would in circular motion. The second you start to confuse speed with velocity is the second that your force calculations will start to fall apart.
Acceleration as Change in Velocity
Acceleration is the change of velocity which may be a change in speed, or a change in direction or both. Thus even at a constant speed, an object or person can still be accelerating, as when rounding a corner.
The car continuing to travel around the curve at a constant 60 mph illustrates acceleration. Although the speed is not increasing or decreasing, the car is accelerating because it is traveling in a curved path. This is an acceleration in the direction of motion, or along the path of travel. As the car's tires are applying a force sideways on the road, the road in turn exerts an upward force on the tires which is the force responsible for the car's acceleration.
Newton's second law, F = ma, relies on understanding acceleration. But what is acceleration? Many people think of acceleration as "speeding up", and while it's true that you are accelerating if you are getting faster, you are also accelerating if you are getting slower. In fact, you are accelerating if the direction of your velocity is changing, period. The acceleration is measured by the rate of change of the velocity vector.
One-Dimensional Motion with Constant Acceleration
Simplest useful model: acceleration doesn't change. Dropping things near Earth's surface is easily described with the constant acceleration (g ≈ 9.8 m/s²), and a car speeding up at a constant throttle is accelerating at a roughly constant rate until drag becomes important. These constant acceleration equations show up over and over.
This is the constant acceleration model that you learned about in your high school physics class. The key difference for engineers is knowing when such a model is appropriate, and when it is not. Constant acceleration is a good approximation for short times, simple systems, and back-of-the-envelope estimates. However it is not accurate for most real world situations where the acceleration is actually changing with time. Understanding this simple concept is important for accurate engineering work.
Two-Dimensional Motion Using Vector Components
Real motion is not in one dimension, it is in two dimensions (2D) or three dimensions (3D). Breaking things into components is the key. Treat the x-direction and y-direction as separate quantities (as long as they are perpendicular to each other, or orthogonal components) and observe that the velocity in the x-direction does not affect the velocity in the y-direction. However, the two components of velocity do combine to form a resultant vector.
Magnitude (speed):
Components are independent and can be combined using the Pythagorean theorem and not addition.
Component thinking applies to anything moving in more than one direction. Analyze projectiles along the horizontal and vertical directions, planar mechanical systems along the x and y (or radial and tangential) directions, and study vibrating systems along their principal components. Learn good component thinking now, or struggle with it later.
Interpreting Motion Graphs for Engineering Analysis
Graphs aren't just pictures—they encode relationships:
Slope of position-time graph = velocity
How fast position changes with time.
Slope of velocity-time graph = acceleration
How fast velocity changes with time.
Area under velocity-time graph = displacement
Accumulate velocity over time, you get total displacement.
Engineers that design systems use graphs to test the reasonableness of models of those systems. Looking at a position versus time plot, for example, they expect to see a straight line (indicating constant velocity, i.e., zero acceleration). A step function (indicating an instantaneous change in velocity, i.e., infinite acceleration) is most definitely not reasonable and indicates that there is a problem with the model. Being able to read graphs like these quickly is a valuable skill.
Task: Describe Motion in 2D (With Full Worked Answer)
A point moves in a plane with initial velocity v0 in direction theta. The object of this simulation is to demonstrate how velocity influences movement, with the trajectory tracing the resultant of linear momentum.
and constant acceleration:
1. Velocity at time t = 3 s.
2. Calculate the speed of the object at t = 3 seconds.
3. Why the speed cannot be found by adding the components?
Solution (Step-by-Step)
If you set up the vector from the initial to the final position, then the length of that vector is the distance traveled.
For each component:
This works because vector equations apply component-by-component.
Multiply the acceleration vector by 3:
Add vectors:
The velocity vector to point 3 is:
Answer (1): v(3) = [12, 5] m/s
Calculate speed from velocity using the formula for the magnitude of a vector.
Speed is the magnitude of the velocity vector:
Substitute components:
Answer (2): speed at 3 s = 13 m/s
Adding components gives:
However, this is not the speed since the components are in perpendicular directions. The correct combination of components to obtain the overall speed is not simply addition, but rather according to the Pythagorean magnitude rule:
Answer (3): Speed is defined as the magnitude of velocity and not the sum of the components. The vector components of a velocity need to be added together with the magnitude of in order to obtain the total velocity.
What You Should Take From Physics Level 1
- 2D motion is treated using vectors, component by component.
- Speed is always the magnitude of velocity
- displacement and velocity are measured as vectors, while distance and speed are measured as scalars.
- Correct vector handling prevents common errors early
Apply Motion and Vector Analysis in Specializations
Vector analysis and kinematics are fundamental to:
- Aerospace & Marine - Analyze flight paths, trajectories, and vehicle dynamics in 3D space
- Mechatronics & Robotics - Design robot motion planning and control systems