Displacement: The Vector of Motion
Distance vs. Displacement: The Core Difference
Imagine you walk from your home to the library and then to a friend's house. The total ground you covered, the twists and turns of your path, is the distance. It only cares about "how much" you moved. Now, imagine drawing a straight arrow from your front door directly to your friend's house. This arrow represents your displacement. It tells you the shortest possible path between your start and end points and the direction you would need to go.
This is the central distinction:
- Distance is a scalar quantity. It has magnitude (a numerical value) but no direction. It is always positive or zero.
- Displacement is a vector quantity. It has both magnitude and a specific direction.
Let's solidify this with a classic example. Suppose a car drives 5 km East from point A to point B, and then 3 km West to point C.
- Total Distance Traveled: 5 km + 3 km = 8 km. This is a simple sum of the path lengths.
- Overall Displacement: Let's define East as the positive direction. The car's initial position is A. After the first leg, its position is +5 km. After moving West (the negative direction), its final position is +5 km - 3 km = +2 km. Therefore, the displacement is +2 km, or 2 km East.
Notice that the displacement is not 8 km; it's the net change in position. If the car drove back to point A, its displacement would be zero, even though the distance traveled would be 10 km.
Describing Displacement as a Vector
Because displacement is a vector, we need more than a number to describe it fully. We need its magnitude and its direction.
Magnitude: This is the length or size of the displacement vector. It is always a positive number and is measured in units of length, such as metres (m). In the car example above, the magnitude of the displacement is 2 km.
Direction: This can be described in many ways, depending on the context:
- Compass directions: North, South, East, West, 30° North of East.
- Relative to a fixed point: Towards the school, away from the origin.
- Using a sign convention: Positive for one direction (e.g., up, right) and negative for the opposite (e.g., down, left).
A displacement of 5 m is incomplete information. A displacement of 5 m, North is a complete description of the vector.
Calculating Displacement in One and Two Dimensions
The method for calculating displacement changes slightly depending on whether the motion is in a straight line (1D) or on a plane (2D).
One-Dimensional Displacement
This is the simplest case, where motion happens along a straight line (e.g., a number line). We use the formula $\vec{s} = \vec{x_f} - \vec{x_i}$ with a defined positive direction. The sign of the result ($+$ or $-$) immediately tells us the direction.
Example: A robot moves along a straight track. It starts at a position of +2 m. It moves to -5 m, and then to +1 m. What is its displacement from the start to the end?
- Initial position, $\vec{x_i} = +2 m$
- Final position, $\vec{x_f} = +1 m$
- Displacement, $\vec{s} = \vec{x_f} - \vec{x_i} = +1 - (+2) = -1 m$
The displacement is -1 m. The magnitude is 1 m, and the negative sign indicates direction (e.g., to the left, if we defined right as positive).
Two-Dimensional Displacement
When movement happens on a flat surface, like a soccer field, we need to consider two perpendicular directions, usually the x-axis and y-axis. Displacement becomes the straight-line path from the starting coordinate ($x_i$, $y_i$) to the final coordinate ($x_f$, $y_f$).
The components of the displacement vector are:
- $s_x = x_f - x_i$ (displacement in the x-direction)
- $s_y = y_f - y_i$ (displacement in the y-direction)
The magnitude of the displacement vector is found using the Pythagorean theorem, as it is the hypotenuse of a right triangle with sides $s_x$ and $s_y$:
The direction is given by the angle $\theta$ the vector makes with a reference axis (usually the positive x-axis), calculated using trigonometry:
Displacement in Action: Real-World Scenarios
Let's apply these concepts to practical situations to see how displacement is used to analyze motion.
Scenario 1: The Hiking Trail
A hiker walks 4 km due North and then 3 km due East.
- Distance: 4 km + 3 km = 7 km.
Displacement: Let's set the starting point as the origin (0, 0). The final position is (3, 4) km.
- Magnitude: $|\vec{s}| = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 km$
- Direction: $\theta = \tan^{-1}\left(\frac{4}{3}\right) \approx 53°$ North of East.
So, the hiker's displacement is 5 km at 53° North of East. This is the "as-the-crow-flies" path from start to finish.
Scenario 2: The Amusement Park Ride
A Ferris wheel has a diameter of 20 m. If you get on at the bottom and the wheel makes one complete revolution, what is your displacement?
- Distance Traveled: The circumference of the wheel, which is $\pi \times diameter = \pi \times 20 m \approx 62.8 m$.
- Displacement: After one full revolution, you are back exactly where you started. Your initial and final positions are identical. Therefore, your displacement is zero.
This scenario perfectly illustrates that displacement depends only on the starting and ending points, not the path taken between them.
| Feature | Distance | Displacement |
|---|---|---|
| Definition | Total path length covered. | Straight-line change in position from start to end. |
| Quantity Type | Scalar | Vector |
| Symbol | $d$ | $\vec{s}$ |
| Direction | No direction. | Has a specific direction. |
| Value | Always positive or zero. | Can be positive, negative, or zero. |
| Path Dependence | Depends on the actual path taken. | Independent of the path taken. |
Common Mistakes and Important Questions
Q: Can displacement be greater than the distance traveled?
A: No. The displacement is the shortest distance between two points. The distance traveled is the length of the actual path, which can never be shorter than the straight-line path. Therefore, the magnitude of displacement is always less than or equal to the total distance traveled. They are only equal if the object moves in a straight line without changing direction.
Q: If displacement is zero, does that mean no motion occurred?
A: Not necessarily. As shown in the Ferris wheel example, an object can travel a long distance and end up back at its starting point, resulting in zero displacement. So, zero displacement only tells you that the initial and final positions are the same; it says nothing about the journey in between. The object could have been moving the entire time.
Q: Why is velocity related to displacement and not distance?
A: Velocity is defined as the rate of change of displacement. It tells you how fast an object's position is changing and in what direction. Speed, on the other hand, is the rate of change of distance. Since displacement is a vector, velocity is also a vector. Using displacement allows us to understand not just how fast something is moving, but also the direction of its motion, which is crucial for fully describing movement.
Footnote
1 Scalar Quantity: A physical quantity that has only magnitude and no direction. Examples include distance, speed, mass, and time.
2 Vector Quantity: A physical quantity that possesses both magnitude and direction. Examples include displacement, velocity, force, and acceleration.
3 Magnitude: The size or length of a vector quantity. It is a scalar and is always a positive value (or zero).
4 Pythagorean Theorem: A fundamental relation in geometry stating that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The formula is $a^2 + b^2 = c^2$.