Distance and Displacement

Ever wonder how your car's speedometer knows how fast you're going? It calculates speed based on distance traveled over time. While speed tells you how fast something moves, it doesn't tell you which direction - making it a scalar quantity (has magnitude but no direction).

Physics often cares about direction, which is where vectors come in. Vectors have both magnitude and direction and are written with a special arrow notation (like $\bar{v}$). Position, displacement, velocity, and acceleration are all vector quantities because they tell us not just "how much" but also "which way."

When describing position, we need a reference frame - a starting point from which all measurements are made. Distance is always positive and represents the total path length traveled. Displacement represents the change in position and has both magnitude and direction, making it a vector that can be positive or negative depending on direction.

Think of it this way: If you walk 3 miles in a circle and end up where you started, your distance is 3 miles, but your displacement is zero - you haven't actually changed position!

Direction

When working with vectors, we need to specify direction. Often we use cardinal directions (North, South, East, West) or angles. For example, a displacement of 3 km [N] tells us you moved 3 kilometers northward.

We can get more specific with angles like "42° W of N" or "19° E of S." When in doubt about directions, always sketch it out! In one-dimensional problems, we typically use East as positive and West as negative.

Let's apply this: If a cyclist rides 3 km west and then 2 km east, the total distance traveled is 5 km. However, the displacement is -1 km (or 1 km [W]), because she ended up 1 km west of her starting point.

Pro tip: Always use $\Delta x$ to indicate displacement and "d" for distance in your calculations. This helps keep these related but different concepts separate in your work.

Position, Distance and Displacement in Practice

Position depends entirely on your chosen reference point. If we set the Cinema as our reference point (0), then the Post Office might be at position 23.4 m [E]. When you walk between locations, your position changes, the distance you travel accumulates, and your displacement represents your direct path from start to finish.

For two-dimensional motion, we need to use the Pythagorean theorem and trigonometry. If you walk 3 m east and then 5 m south, your total distance is 8 m, but your displacement is $\sqrt{3^2 + 5^2} = \sqrt{34}$ m (about 5.8 m) at an angle of about 59° south of east.

Working with angles requires breaking vectors into components. For example, if a paper blows 18 m at 30° east of south, we can find:

• East component: 18 m × sin(60°) = 15.6 m
• South component: 18 m × cos(60°) = 9 m

Remember: When vectors move in more than one direction, always draw a right triangle and use SOHCAHTOA to find the components. Sketching vectors (adding tip to tail) makes these problems much easier to solve!