Motion and Vectors
(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.
Scalars vs Vectors
A scalar is just a number: 5 meters, 10 seconds, 50 kg. Magnitude only. A vector has magnitude and direction: 5 meters east, 10 m/s upward, 50 N to the right. Direction matters in engineering because forces, velocities, and accelerations don't just happen—they happen in specific directions.
Vector: Velocity = [3, −4] m/s (3 m/s in x-direction, −4 m/s in y-direction)
The vector tells you where it's going. The scalar doesn't.
Mess up the direction and your calculation might be mathematically correct but physically meaningless. Forces, momentum, torque, fluid flow—all vectors. If you're not comfortable with vector thinking, you'll struggle with everything that follows.
Position, Displacement, and Distance
Position is where you are (relative to some origin). Displacement is how far you've moved from start to finish (a vector—includes direction). Distance is how much ground you covered along the way (a scalar—no direction, just total path length).
• Distance traveled = 6 meters (you moved 6 meters total)
• Displacement = 0 (you ended where you started)
Why This Matters Later
Displacement determines how systems change state. You can travel 100 miles and end up exactly where you started—zero displacement. For physics, that's fundamentally different from traveling 100 miles in a straight line. Distance alone doesn't tell you direction, so it can't tell you how position changed.
Velocity and Speed
Velocity is the rate of change of position. It's a vector: tells you how fast and in what direction. Speed is just the magnitude of velocity—how fast you're going, period. No direction info.
You're moving 4 m/s in the x-direction and 3 m/s in the y-direction.
Speed:
Speed is always the length of the velocity vector. Doesn't matter which direction you're going—5 m/s is 5 m/s.
This distinction shows up constantly. When you analyze forces, you care about how velocity changes—both magnitude and direction. You can have constant speed but changing velocity (think circular motion: speed stays the same, but direction keeps changing, so velocity is changing). Get this wrong and force calculations fall apart.
Acceleration
Acceleration is how velocity changes over time. That includes speeding up, slowing down, or turning. Even if your speed stays constant, if you're changing direction, you're accelerating.
Example: Car going around a curve at constant 60 mph. Speed's not changing, but direction is. That's acceleration. The tires push sideways on the road, the road pushes back, and that force causes the acceleration (change in velocity direction).
This is where Newton's second law comes in (F = ma). Force causes acceleration. But you can't apply F = ma correctly if you don't understand what acceleration actually is. Changing speed? Acceleration. Changing direction? Also acceleration. Both at once? Still acceleration. It's all about how the velocity vector changes.
Motion in 1D: Constant Acceleration
Simplest useful model: acceleration doesn't change. Dropping things near Earth's surface? Constant acceleration (g ≈ 9.8 m/s²). Car speeding up at steady throttle? Roughly constant acceleration (until drag becomes significant). These equations show up everywhere:
You've seen these in physics class. The difference in engineering is knowing when they apply and when they don't. Constant acceleration is an approximation. It works great for short durations, simple systems, and back-of-the-envelope estimates. It breaks down when acceleration changes with time (most real systems). Know the limits.
Motion in 2D: Vector Components
Real motion happens in 2D or 3D, not along a single axis. The trick: break everything into components. Treat x-direction and y-direction independently (as long as they're perpendicular—orthogonal components). Motion in x doesn't affect motion in y. They recombine to give you the full vector.
Magnitude (speed):
Components are independent. Combine them using Pythagorean theorem, not addition.
Component-based thinking is how you analyze anything that moves in more than one dimension. Projectile motion? Separate horizontal and vertical. Mechanisms? Decompose into x and y (or radial and tangential). Vibration modes? Same idea, different context. Master this now or struggle later.
Motion Graphs: What They Actually Mean
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 use graphs to sanity-check models. If your simulation shows position increasing linearly with time, you know velocity is constant (zero acceleration). If velocity jumps discontinuously, you've got infinite acceleration (unphysical—something's wrong with your model). Learn to read graphs quickly and you catch errors before they propagate.
Task: Describe Motion in 2D (With Full Worked Answer)
A point moves in a plane with initial velocity:
and constant acceleration:
1. Find the velocity vector after t = 3 seconds.
2. Find the speed at t = 3 seconds.
3. Explain why the speed is not found by adding components.
Solution (Step-by-Step)
For each component:
This works because vector equations apply component-by-component.
Multiply the acceleration vector by 3:
Add vectors:
Answer (1): v(3) = [12, 5] m/s
Speed is the magnitude of the velocity vector:
Substitute components:
Answer (2): speed at 3 s = 13 m/s
Adding components gives:
but this is not the speed because components are perpendicular directions. The correct combination is not addition; it is the Pythagorean magnitude rule:
Answer (3): Speed is the length of the velocity vector, not the sum of its components. Components must be combined using magnitude because they represent orthogonal directions.
What You Should Take From This Task
- Motion in 2D is handled by vectors, component-by-component
- Speed is always the magnitude of velocity
- Correct vector handling prevents common errors early
Ready for the Next Level?
Once you understand how to describe motion with vectors, you're ready to explore what causes that motion.
Continue to Level 2: Forces and Energy →