Displacement Equation- Calculating Motion in Physics
What Is Displacement in Physics?
Displacement is the shortest straight-line distance between two points. It's not the same as distance traveled. If you walk 10 meters east and then 10 meters west, your distance is 20 meters but your displacement is zero.
Physics problems care about displacement because it includes direction. That's why displacement is a vector quantity — it has both magnitude and direction. Distance, by contrast, is just a number (scalar).
The Basic Displacement Equation
The most fundamental displacement equation comes from average velocity:
s = v × t
Where:
- s = displacement (meters)
- v = average velocity (meters/second)
- t = time (seconds)
This only works when velocity is constant. Most real physics problems involve acceleration, which means you need the full kinematic equations.
The Four Kinematic Equations for Displacement
These are the equations you'll actually use in physics class. They relate displacement, velocity, acceleration, and time.
Equation 1: Displacement with Constant Acceleration
s = ut + ½at²
This is your go-to equation when you know initial velocity, acceleration, and time — but not final velocity.
Equation 2: Displacement Without Time
v² = u² + 2as
Rearranged for displacement: s = (v² - u²) / 2a
Use this when time isn't given. It's incredibly useful for projectile motion problems.
Equation 3: Displacement Using Average Velocity
s = ½(u + v)t
Average velocity equals (initial + final velocity) / 2. This only applies when acceleration is constant.
Equation 4: Alternative Form
s = vt - ½at²
Same as equation 1 but rearranged. Some problems give you final velocity instead of initial velocity — this is the version to use.
Quick Reference: Kinematic Equations Table
| What You Know | Use This Equation | Form |
|---|---|---|
| u, a, t | Equation 1 | s = ut + ½at² |
| u, v, a | Equation 2 | s = (v² - u²) / 2a |
| u, v, t | Equation 3 | s = ½(u + v)t |
| v, a, t | Equation 4 | s = vt - ½at² |
u = initial velocity, v = final velocity, a = acceleration, t = time, s = displacement
How to Calculate Displacement: Getting Started
Here's the process that actually works:
Step 1: Identify Known Variables
Read the problem. What information is given? Initial velocity? Final velocity? Acceleration? Time? Write down what you have.
Step 2: Identify Unknown Variables
What does the problem ask you to find? Displacement? Time? Acceleration? Be specific.
Step 3: Choose the Right Equation
Match your knowns to the equation that contains your unknown. If you have u, a, and t — use s = ut + ½at². Don't force an equation that doesn't fit.
Step 4: Plug In the Numbers
Substitute your values. Watch your units — everything must be consistent. Convert km/h to m/s if needed (divide by 3.6).
Step 5: Solve
Do the algebra. Isolate the variable. Calculate. Check that your answer makes sense.
Worked Example
Problem: A car accelerates from rest (u = 0) at 4 m/s² for 6 seconds. What is its displacement?
Solution:
Known: u = 0, a = 4 m/s², t = 6 s
Unknown: s
Equation: s = ut + ½at²
s = (0)(6) + ½(4)(6)²
s = 0 + ½(4)(36)
s = 2 × 36
s = 72 meters
Vector Displacement in 2D
When motion happens in two dimensions, you need to break it into components. The displacement vector has x and y parts:
s = √(sx² + sy²)
Where sx and sy are the horizontal and vertical displacement components. The direction (angle) is:
θ = tan⁻¹(sy / sx)
Common Mistakes That Ruin Your Answers
- Confusing distance with displacement — they're not the same. Displacement is the straight line between start and end points.
- Forgetting that displacement is a vector — direction matters. A displacement of +10m is different from -10m.
- Using the wrong equation — match your known variables to the right formula. Randomly picking equations wastes time.
- Unit conversion errors — km/h to m/s is divide by 3.6. m/s to km/h is multiply by 3.6. Getting this wrong poisons everything.
- Not checking if acceleration is constant — kinematic equations only work with constant acceleration.
Displacement vs. Distance: The Bottom Line
Displacement measures change in position. It gives you the net result, not the path taken. In physics problems, always read carefully — if the question asks for displacement, use vector equations. If it asks for distance traveled, you may need to add up segments of the path.
The equations above cover 95% of what you'll encounter in introductory physics. Master these four kinematic equations and you can solve essentially any basic displacement problem thrown at you.