How to Find Displacement- Physics Problems Solved
What Displacement Actually Is (And Why Your Textbook Is Confusing You)
Displacement is the shortest distance between two points in a straight line. That's it. Not the path traveled—those are two completely different things and your professor will test you on the difference.
Direction matters. Displacement is a vector, which means it has both magnitude (how far) and direction (which way). If you walk 10 meters east then 10 meters west, your distance traveled is 20 meters. Your displacement is zero. You ended up where you started.
The Displacement Formula You Actually Need
Here's the equation:
Δx = x₂ - x₁
Where:
- Δx = displacement (the change in position)
- x₂ = final position
- x₁ = initial position
The triangle symbol (Δ) means "change in." So you're just subtracting where you started from where you ended up.
For Motion With Constant Velocity
If you're dealing with velocity and time:
x = x₀ + v·t
Where:
- x = final position
- x₀ = initial position
- v = velocity
- t = time elapsed
How to Find Displacement: Step-by-Step
Let's say you have a problem. Here's how you actually solve it:
- Identify your initial position (x₁) — where does the object start?
- Identify your final position (x₂) — where does it end up?
- Subtract — Δx = x₂ - x₁
- Add direction — state whether it's east/west, positive/negative, or along an axis
Example Problem
A car starts at position x = 5 meters. It drives to x = 47 meters. What is the displacement?
Solution:
Δx = x₂ - x₁
Δx = 47 - 5
Δx = 42 meters (in the positive direction)
Displacement vs. Distance: The Table Your Textbook Won't Show You
| Feature | Displacement | Distance |
|---|---|---|
| Type | Vector (has direction) | Scalar (no direction) |
| Can be negative? | Yes | No |
| Always ≤ distance? | Yes | No (distance is always ≥ displacement) |
| Path matters? | No | Yes |
| Can be zero? | Yes (if you return to start) | Yes (if you return to start) |
| Shortest path? | Always the shortest | Depends on the path taken |
Common Mistakes That Cost You Points
These errors show up constantly in homework and exams:
- Confusing displacement with distance traveled — Read the question. If it asks for displacement, you need the straight-line answer, not the total path.
- Forgetting to include direction — Your answer isn't complete without stating the direction. "42 meters" is half an answer. "42 meters east" is correct.
- Getting initial and final positions backwards — x₂ is always final, x₁ is always initial. Don't swap them.
- Using the wrong sign — If you define right as positive, then left is negative. Be consistent with your coordinate system.
- Mixing up displacement with velocity — Displacement is position change. Velocity is how fast that change happens. Different concepts.
Displacement in 2D: When Things Get Real
Most students handle 1D problems fine, then panic when they see 2D motion. Here's what changes:
You now have x and y components. The displacement vector has two parts:
Δr = √(Δx² + Δy²)
Where Δx is horizontal displacement and Δy is vertical displacement.
2D Example
A drone flies 30 meters north, then 40 meters east. What's its total displacement from the starting point?
Δx = 40 m (east)
Δy = 30 m (north)
Δr = √(40² + 30²) = √(1600 + 900) = √2500 = 50 meters
The direction? Use trigonometry: tan(θ) = Δy/Δx = 30/40, so θ = 36.9° north of east.
Velocity-Time Graphs and Displacement
The area under a velocity-time graph is displacement. Here's why that matters:
- When velocity is positive, the area adds to displacement
- When velocity is negative, the area subtracts from displacement
- Count the squares, multiply by the scale factor, and you get displacement without doing kinematics calculations
This trick works for any shape—just break it into shapes you can calculate (rectangles, triangles) and add them up.
Quick Reference: Formulas That Actually Come Up
| Situation | Formula |
|---|---|
| Basic displacement | Δx = x₂ - x₁ |
| Constant velocity | x = x₀ + vt |
| Average velocity | v_avg = Δx / Δt |
| 2D displacement | |Δr| = √(Δx² + Δy²) |
| From acceleration | x = x₀ + v₀t + ½at² |
When Displacement Equals Zero
This specific case trips up a lot of people. Displacement is zero when:
- An object returns to its starting point
- The initial and final positions are identical
But distance traveled can still be huge. A satellite orbiting Earth has zero displacement after one complete orbit—but it's traveled roughly 40,000 kilometers. The question is testing whether you understand the difference.
Getting Started: Your First 5 Problems
Don't read more. Go solve problems. Here's your practice setup:
- Find displacement: starts at x = -3m, ends at x = 7m
- Find displacement: starts at x = 15m, ends at x = -5m
- A bird flies 100m north, then 100m south. What's the displacement?
- A runner completes a 400m circular track. What's the displacement?
- A particle moves from (2, 3) to (5, 7). Find the displacement vector.
Check your answers: #1 = 10m, #2 = -20m, #3 = 0m, #4 = 0m, #5 = (3, 4) with magnitude 5.