Kinematics in One Dimension
Kinematics is the study of how objects move along a straight line, without considering the forces that cause the motion. It answers questions like: How far? How fast? How quickly is the speed changing?
This guide covers position, displacement, velocity, acceleration, the four kinematic equations, free fall, worked examples, key formulas, and a 10-question practice quiz.
Kinematics: Motion in One Dimension
1What Is Kinematics and Why Does It Matter?
Kinematics in one dimension is the study of how objects move along a straight line, without considering the forces that cause the motion. It is fundamental to understanding more complex movements in engineering, sports, technology, and daily life.
Kinematics describes motion using three primary quantities: displacement, velocity, and acceleration. By understanding how these quantities relate to each other, we can predict an object's future position or velocity, or deduce its past motion.
A car accelerates from a traffic light. You feel pushed back into your seat. The speedometer climbs from 0 to 60 mph. This is kinematics in action — describing how things move without asking why they speed up or slow down.
- Engineering — designing vehicles, rollercoasters, and automated systems
- Sports — analyzing a sprinter's start or a ball's trajectory
- Driving — calculating stopping distances and travel times
- Free fall — predicting how long a dropped object takes to hit the ground
2Key Definitions
Position (x)
Location of an object relative to a chosen origin. Units: meters (m)
Displacement (Δx)
Change in position. A vector quantity (has direction). Δx = x₂ − x₁. Units: m
Distance (d)
Total path length traveled. A scalar quantity (only magnitude, always positive). Units: m
Speed (v)
How fast an object moves. A scalar. Units: m/s
Velocity (v)
Speed with direction. A vector. In 1D, sign indicates direction. Units: m/s
Acceleration (a)
Rate of change of velocity. A vector. Units: m/s²
Initial Velocity (v₀)
Velocity at t = 0 (v-naught). Units: m/s
Free Fall
Motion under gravity alone (no air resistance). Acceleration = g ≈ 9.8 m/s² downward.
3Position, Displacement & Distance
Imagine a number line. Position (x) is a point on that line. Displacement (Δx) is the change in position — it is a vector with magnitude and direction. Distance (d) is the total path length traveled — always positive.
Δx = xfinal − xinitial
Δx = displacement (m) — can be positive or negative
d = distance (m) — always positive (total path length)
Walk 5 m forward and 3 m backward. Your displacement is +2 m, but your distance traveled is 8 m. Always define a positive direction and stay consistent!
4Velocity
Velocity describes how fast an object's position changes and in what direction. Unlike speed (a scalar), velocity is a vector.
vavg = Δx / Δt
vavg = average velocity (m/s)
Δx = displacement (m)
Δt = time interval (s)
Key Concepts
5Acceleration
Acceleration is the rate at which velocity changes. It's a vector quantity with units of m/s².
a = Δv / Δt
a = acceleration (m/s²)
Δv = change in velocity (m/s)
Δt = time interval (s)
Interpreting Signs of Acceleration
Speeding Up
Velocity and acceleration have the same sign. (Both positive or both negative.)
Slowing Down (Deceleration)
Velocity and acceleration have opposite signs. (One positive, one negative.)
Graph Interpretation
6Kinematic Equations for Constant Acceleration
These four equations link displacement (Δx), initial velocity (v₀), final velocity (v), acceleration (a), and time (t). They apply only when acceleration is constant.
Equation 1 (no Δx)
v = v₀ + at
Equation 2 (no v)
Δx = v₀t + ½at²
Equation 3 (no t)
v² = v₀² + 2aΔx
Equation 4 (no a)
Δx = ((v₀ + v) / 2)t
- List your knowns (v₀, v, a, Δx, t)
- Identify your unknown (what you need to find)
- Select the equation that omits the variable you don't know and don't need
7Free Fall
Free fall is a special case of constant-acceleration motion where the only force is gravity. All objects fall at the same rate regardless of mass (ignoring air resistance).
g ≈ 9.8 m/s²
If up is positive: a = −9.8 m/s² (always downward)
Thrown Upward
v₀ > 0, slows down, stops at peak (v = 0), then falls back
At Maximum Height
v = 0 m/s momentarily, but a = −9.8 m/s² still acts
Dropped Object
v₀ = 0, speeds up at 9.8 m/s every second
Free fall is simply constant-acceleration motion with a = −9.8 m/s². Use the same four kinematic equations — just replace a with −g and Δx with Δy.
8Worked Examples
Example 1: Runner's Average Velocity (Basic)
A runner travels 100 meters east in 12 seconds. What is their average velocity?
Given: Δx = 100 m, Δt = 12 s
Formula: vavg = Δx / Δt
Step 1: Substitute: vavg = 100 m / 12 s
Step 2: Calculate: vavg ≈ 8.33 m/s East
Example 2: Car Accelerating from Rest (Basic)
A car starts from rest and accelerates uniformly at 3.0 m/s² for 4.0 s. What is its final velocity?
Given: v₀ = 0 m/s, a = 3.0 m/s², t = 4.0 s
Formula: v = v₀ + at
Step 1: Substitute: v = 0 + (3.0)(4.0)
Step 2: Calculate: v = 12.0 m/s
Example 3: Bicycle Displacement (Intermediate)
A bicycle starts at 5.0 m/s and accelerates at 2.0 m/s² for 10 s. How far does it travel?
Given: v₀ = 5.0 m/s, a = 2.0 m/s², t = 10 s
Formula: Δx = v₀t + ½at²
Step 1: Δx = (5.0)(10) + ½(2.0)(10)² = 50 + 100
Step 2: Δx = 150 m
Example 4: Stone Dropped from Cliff (Intermediate)
A stone is dropped from a cliff. What is its velocity after 3.0 s? (g = 9.8 m/s²)
Given: v₀ = 0 m/s, t = 3.0 s, a = −9.8 m/s² (up is positive)
Formula: v = v₀ + at
Step 1: v = 0 + (−9.8)(3.0)
Step 2: v = −29.4 m/s (29.4 m/s downward)
Example 5: Braking Car (Advanced)
A car traveling at 20 m/s brakes and stops in 50 m. Find its acceleration and stopping time.
Part A — Find a: v² = v₀² + 2aΔx
Step 1: 0 = (20)² + 2a(50) → 0 = 400 + 100a
Step 2: a = −4.0 m/s²
Part B — Find t: v = v₀ + at
Step 3: 0 = 20 + (−4.0)t → t = 5.0 s
9Key Formulas at a Glance
Avg. Velocity
v = Δx/Δt
Avg. Acceleration
a = Δv/Δt
Equation 1
v = v₀ + at
Equation 2
Δx = v₀t + ½at²
Equation 3
v² = v₀² + 2aΔx
Equation 4
Δx = ((v₀+v)/2)t
| Formula | Missing Variable | When to Use |
|---|---|---|
| v = v₀ + at | Δx | Don't know/need displacement |
| Δx = v₀t + ½at² | v | Don't know/need final velocity |
| v² = v₀² + 2aΔx | t | Don't know/need time |
| Δx = ((v₀ + v)/2)t | a | Don't know/need acceleration |
| g ≈ 9.8 m/s² | — | Free-fall problems (a = −g if up is +) |
10Memory Aids
S = Displacement, U = Initial Velocity, V = Final Velocity, A = Acceleration, T = Time. Each kinematic equation is missing exactly one SUVAT variable — pick the equation that drops what you don't need.
Always remember: g = 9.8 m/s² is a magnitude. When up is positive, use a = −9.8 m/s² in your equations. Stick with "up is positive" for consistency.
List the 5 variables (v₀, v, a, Δx, t). You'll be given three, asked for one, and one is irrelevant. Pick the equation that doesn't contain the irrelevant one.
x-t slope = velocity. v-t slope = acceleration. v-t area = displacement. Remember: slope goes "up" one level, area goes "down" one level.
Don't overcomplicate it! Free fall just means a = −9.8 m/s². Use the same kinematic equations. Replace Δx with Δy and you're done.
11Common Mistakes Students Make
"Negative acceleration always means slowing down."
Not true! If the object is moving in the negative direction and has negative acceleration, it is speeding up. Deceleration occurs when velocity and acceleration have opposite signs.
"Forgetting to convert units before calculating."
Always check that all quantities are in consistent SI units (meters, seconds, m/s, m/s²). If given km/h or cm, convert to m/s or m before plugging into formulas.
"Confusing distance and displacement."
Displacement (Δx) is a vector — it can be negative. Distance is a scalar — always positive. Walking 5 m forward and 5 m back: Δx = 0 m, but distance = 10 m.
"Assuming initial velocity is always zero."
Many problems state "starts from rest" (v₀ = 0), but not all! If an object is already moving, its v₀ will be a non-zero value. Read the problem carefully.
"Using g = +9.8 when up is positive."
If you set "up" as your positive direction, then acceleration due to gravity must be a = −9.8 m/s². A positive a would imply gravity pushes the object up.
"Picking the wrong kinematic equation."
Identify which variable is missing/irrelevant and choose the equation that omits it. Don't pick at random — each equation is designed to exclude one specific variable.
Frequently Asked Questions
- What's the difference between speed and velocity?
- Speed is a scalar quantity that tells you how fast an object is moving (e.g., 50 km/h). Velocity is a vector quantity that tells you both how fast an object is moving AND in what direction (e.g., 50 km/h North). In one dimension, the sign (+ or -) indicates direction for velocity.
- Does negative acceleration always mean an object is slowing down?
- No. Negative acceleration means the acceleration vector points in the negative direction. If the object is moving in the positive direction (positive velocity), then negative acceleration will cause it to slow down (decelerate). However, if the object is already moving in the negative direction (negative velocity), then negative acceleration will cause it to speed up in the negative direction.
- When do I use g = 9.8 m/s² and when do I use a = -9.8 m/s²?
- g itself is defined as the magnitude of the acceleration due to gravity, which is 9.8 m/s². When you use it in the kinematic equations for a, you must assign a sign based on your chosen coordinate system. If you define "up" as positive, then gravity acts downwards, so a = -9.8 m/s². If you define "down" as positive, then a = +9.8 m/s². It's generally recommended to consistently choose "up" as positive.
- Can an object have zero velocity but non-zero acceleration?
- Yes! The classic example is an object thrown vertically upwards. At the very peak of its trajectory, its instantaneous velocity is 0 m/s for a brief moment as it changes direction. However, the acceleration due to gravity (-9.8 m/s²) is still acting on it, continuously. If the acceleration were also zero, the object would just hang motionless in the air.
- How do I know which kinematic equation to use?
- For any problem, list out the five kinematic variables: vv₀, v, a, Δx, t. You'll typically be given three of these, asked to find one, and one variable will be neither given nor asked for. Choose the kinematic equation that does not include the variable you don't know and don't need to find.
Practice Quiz
Test your understanding of kinematics in one dimension — select the correct answer for each question.
1.Which of the following is a vector quantity?
2.A car travels 50 m North, then turns around and travels 20 m South. What is the car's total displacement?
3.The slope of a position-time graph gives:
4.An object is moving with a constant velocity. Its acceleration is:
5.A ball is thrown straight up. At its maximum height, its velocity is:
6.A car accelerates from 10 m/s to 20 m/s in 5 s. What is its average acceleration?
7.Which kinematic equation would you use if you did NOT know the final velocity (v) and did NOT need to find it?
8.An object in free fall near the Earth's surface experiences an acceleration of approximately 9.8 m/s². If "up" is considered the positive direction, what value should be used for acceleration "a" in the kinematic equations?
9.The area under a velocity-time graph represents:
10.A car starts from rest and accelerates at 4.0 m/s². How far does it travel in 3.0 s?
Final Study Advice
- 1. Always define a positive direction before solving any kinematics problem.
- 2. List all knowns and unknowns. Identify which variable is missing/irrelevant to choose the right equation.
- 3. Draw a diagram — label positions, velocities, and acceleration with signs.
- 4. Convert all units to SI (meters, seconds) before plugging into equations.
- 5. Check your answer: does the sign make sense? Is the magnitude reasonable?