⚡ Velocity Calculator
This velocity calculator solves five different velocity formulas — basic v = d/t, final velocity from acceleration, velocity without time using vf² = vi² + 2ad, average velocity, and velocity from kinetic energy. Every calculation shows full step-by-step working and displays results in m/s, km/h, mph, ft/s, and knots simultaneously — no unit selection required.
Key Formulas:
v = d/t
vf = vi + at
vf² = vi² + 2ad
v_avg = (vi+vf)/2
v = √(2KE/m)
🔵 Basic Velocity — v = d/t
Quick Fill — Famous Speeds
🟢 Final Velocity — vf = vi + at
Formula: vf = vi + at — Use when you know initial velocity, acceleration, and time. In UK notation: v = u + at. The variable vf means the velocity at the end of the time interval.
🟡 Final Velocity (no time) — vf² = vi² + 2ad
Formula: vf² = vi² + 2ad — Use when time is unknown. You need initial velocity, acceleration, and displacement. In UK notation: v² = u² + 2as. This is the third kinematic equation.
🟠 Average Velocity — v_avg = (vi + vf) / 2
⚠️ Important: Average velocity uses displacement (not distance). For a round trip, average velocity = 0. Average speed uses total distance. The formula v_avg = (vi + vf)/2 only works for constant acceleration.
For displacement calculation
🔴 Velocity from Kinetic Energy — v = √(2KE/m)
Formula: KE = ½mv² → v = √(2KE/m) — Rearranging the kinetic energy equation. Given mass and kinetic energy, find velocity. Or given velocity and mass, find kinetic energy.
Velocity Formulas — All Five Equations
Choose the formula based on what you know. The basic formula v = d/t handles constant speed. For acceleration use vf = vi + at or vf² = vi² + 2ad.
| Formula Name | Equation | Use When | Solve For |
|---|---|---|---|
| Basic velocity | v = d/t | Know distance and time | v, d, or t |
| Final velocity (with time) | vf = vi + at | Know acceleration and time | vf, vi, a, or t |
| Final velocity (no time) | vf² = vi² + 2ad | Know acceleration and displacement | vf, vi, a, or d |
| Average velocity (constant a) | v_avg = (vi + vf)/2 | Constant acceleration only | Average speed |
| Average velocity (general) | v_avg = d_total/t_total | Any motion | Average speed |
| Velocity from KE | v = √(2KE/m) | Know kinetic energy and mass | v or KE |
v = d/t
vf = vi + at
vf² = vi² + 2ad
v_avg = (vi+vf)/2
📐 UK vs US notation: In UK physics, initial velocity is written as u and final velocity as v, giving the three kinematic equations as: v = u + at, s = ut + ½at², v² = u² + 2as. In US notation: vf = vi + at, d = vi·t + ½at², vf² = vi² + 2ad. This calculator uses vi and vf throughout, but both notations mean exactly the same thing.
How to Calculate Velocity — Step-by-Step
Step 1: Identify what you know — distance? time? acceleration? initial velocity?
Step 2: Choose the correct formula from the table above.
Step 3: Convert all values to SI (metres and seconds).
Step 4: Substitute values and solve algebraically.
Step 5: Convert the answer to desired output units.
Example 1 — Basic v = d/t:
A car travels 150 km in 1.5 hours.
v = d/t = 150 km / 1.5 h = 100 km/h = 27.78 m/s = 62.14 mph
Check: 27.78 × 5,400 s = 150,012 m ≈ 150 km ✓
A car travels 150 km in 1.5 hours.
v = d/t = 150 km / 1.5 h = 100 km/h = 27.78 m/s = 62.14 mph
Check: 27.78 × 5,400 s = 150,012 m ≈ 150 km ✓
Example 2 — Final velocity vf = vi + at:
A ball drops from rest for 4 seconds. vi = 0, a = 9.81 m/s², t = 4 s.
vf = vi + at = 0 + 9.81 × 4 = 39.24 m/s = 141.3 km/h = 87.8 mph
A ball drops from rest for 4 seconds. vi = 0, a = 9.81 m/s², t = 4 s.
vf = vi + at = 0 + 9.81 × 4 = 39.24 m/s = 141.3 km/h = 87.8 mph
Example 3 — Braking (no time): vf² = vi² + 2ad:
Car at 60 mph brakes to stop over 40 m. vi = 26.82 m/s, vf = 0, d = 40 m.
a = (0² − 26.82²)/(2×40) = −719.3/80 = −8.99 m/s² ≈ −0.916 g
Car at 60 mph brakes to stop over 40 m. vi = 26.82 m/s, vf = 0, d = 40 m.
a = (0² − 26.82²)/(2×40) = −719.3/80 = −8.99 m/s² ≈ −0.916 g
How to Find Final Velocity (vf) — Two Methods
Method 1 — When you know time
vf = vi + at
- vi = initial velocity
- a = acceleration (positive = speeding up)
- t = time elapsed
Example: Rocket: vi=0, a=50 m/s², t=10 s
vf = 0 + 50×10 = 500 m/s = 1,800 km/h = Mach 1.46
vf = 0 + 50×10 = 500 m/s = 1,800 km/h = Mach 1.46
Method 2 — When time is unknown
vf² = vi² + 2ad
- Rearrange: vf = √(vi² + 2ad)
- Use displacement instead of time
- vf must be ≥ 0 (check sign of vi²+2ad)
Example: Roller coaster: vi=5 m/s, a=8 m/s², d=30 m
vf = √(25+480) = √505 = 22.47 m/s = 80.9 km/h
vf = √(25+480) = √505 = 22.47 m/s = 80.9 km/h
| Special Case | Condition | Result |
|---|---|---|
| Constant velocity | a = 0 | vf = vi (no change) |
| Starts from rest | vi = 0 | vf = at (Method 1) or vf = √(2ad) (Method 2) |
| Braking to stop | vf = 0 | t = vi/a and stopping distance d = vi²/(2|a|) |
| Free fall | vi=0, a=9.81 | vf = 9.81t or vf = √(2×9.81×h) |
Average Velocity vs Average Speed — The Key Difference
Average Speed (Scalar)
= total distance / total time
Always positive. Uses total path length.
Average Velocity (Vector)
= total displacement / total time
Can be zero or negative. Uses net displacement.
Classic round trip example: A→B (100 km at 100 km/h) then B→A (100 km at 80 km/h)
Time going: 1.00 h | Time returning: 1.25 h | Total time: 2.25 h
Average speed = 200 km / 2.25 h = 88.9 km/h
Average velocity = 0 km / 2.25 h = 0 km/h (back at start!)
⚠️ The simple average (100+80)/2 = 90 km/h is WRONG — you spend more time at the slower speed.
Time going: 1.00 h | Time returning: 1.25 h | Total time: 2.25 h
Average speed = 200 km / 2.25 h = 88.9 km/h
Average velocity = 0 km / 2.25 h = 0 km/h (back at start!)
⚠️ The simple average (100+80)/2 = 90 km/h is WRONG — you spend more time at the slower speed.
Harmonic mean formula for round-trip average speed:
v_avg = 2v₁v₂/(v₁+v₂) = 2×100×80/(100+80) = 16,000/180 = 88.9 km/h ✓
Velocity vs Speed — What is the Difference
⚡ Speed (Scalar)
- Magnitude only — no direction
- Always positive or zero
- e.g. "60 km/h"
- An odometer measures speed
🧭 Velocity (Vector)
- Magnitude AND direction
- Can be negative (opposite direction)
- e.g. "60 km/h north"
- GPS gives velocity
Key examples:
🎯 Ball thrown up at 10 m/s: speed=10 m/s, velocity=+10 m/s (up).
On return at same speed: velocity=−10 m/s (downward). Speed unchanged, velocity reversed.
🛰️ Satellite orbiting at 7,800 m/s: constant speed, constantly changing velocity (direction changes).
🎯 Ball thrown up at 10 m/s: speed=10 m/s, velocity=+10 m/s (up).
On return at same speed: velocity=−10 m/s (downward). Speed unchanged, velocity reversed.
🛰️ Satellite orbiting at 7,800 m/s: constant speed, constantly changing velocity (direction changes).
Units of speed and velocity are identical (m/s, km/h, mph, ft/s) — but velocity requires a direction to be fully specified.
Velocity Unit Converter — Quick Reference Table
| Unit | m/s | km/h | mph | ft/s | knots |
|---|---|---|---|---|---|
| 1 m/s | 1 | 3.600 | 2.237 | 3.281 | 1.944 |
| 1 km/h | 0.2778 | 1 | 0.6214 | 0.9113 | 0.5400 |
| 1 mph | 0.4470 | 1.6093 | 1 | 1.4667 | 0.8690 |
| 1 ft/s | 0.3048 | 1.0973 | 0.6818 | 1 | 0.5925 |
| 1 knot | 0.5144 | 1.8520 | 1.1508 | 1.6878 | 1 |
| Object / Event | Speed | m/s | km/h | mph |
|---|---|---|---|---|
| Human walking | Typical | 1.4 | 5.0 | 3.1 |
| Usain Bolt (100m WR) | Peak | 12.4 | 44.7 | 27.8 |
| Cyclist (Tour de France) | Peak | 16 | 58 | 36 |
| Cheetah | Top | 29.1 | 105 | 65 |
| Car (motorway) | Typical | 33.3 | 120 | 74.6 |
| Sound (sea level) | Standard | 343 | 1,235 | 767 |
| Passenger jet | Cruising | 250 | 900 | 560 |
| SR-71 Blackbird | Top | 981 | 3,540 | 2,200 |
| Orbital velocity (LEO) | Min orbit | 7,800 | 28,080 | 17,450 |
| Light in vacuum | Maximum | 299,792,458 | 1.08×10⁹ | 6.7×10⁸ |
Initial Velocity Formula — How to Find vi
Initial velocity (vi) — also written as u (UK notation) or v₀ — is the velocity of an object at the START of a time interval. It can be zero (starts from rest) or any value. The initial velocity formula depends on what other quantities you know.
vi = vf − at
vi = √(vf²−2ad)
vi = 2v_avg − vf
vi = d/t − ½at
Formula 1: vi = vf − at (rearranged from vf = vi + at)
Use when: know final velocity, acceleration, and time.
Example: car reaches 30 m/s after 5 s at 4 m/s² → vi = 30 − (4×5) = 10 m/s
Use when: know final velocity, acceleration, and time.
Example: car reaches 30 m/s after 5 s at 4 m/s² → vi = 30 − (4×5) = 10 m/s
Formula 2: vi = √(vf² − 2ad) (rearranged from vf² = vi² + 2ad)
Use when: know final velocity, acceleration, displacement (no time).
Example: ball reaches 20 m/s over 15 m at 10 m/s² → vi = √(400−300) = √100 = 10 m/s
Use when: know final velocity, acceleration, displacement (no time).
Example: ball reaches 20 m/s over 15 m at 10 m/s² → vi = √(400−300) = √100 = 10 m/s
Formula 3: vi = 2×v_avg − vf (from v_avg = (vi+vf)/2)
Use when: know average velocity and final velocity.
Example: v_avg=15 m/s, vf=25 m/s → vi = 30−25 = 5 m/s
Use when: know average velocity and final velocity.
Example: v_avg=15 m/s, vf=25 m/s → vi = 30−25 = 5 m/s
Formula 4: vi = d/t − ½at (from d = vi·t + ½at²)
Use when: know displacement, time, acceleration.
Example: 50 m in 4 s, a=2 m/s² → vi = 50/4 − ½×2×4 = 12.5−4 = 8.5 m/s
Use when: know displacement, time, acceleration.
Example: 50 m in 4 s, a=2 m/s² → vi = 50/4 − ½×2×4 = 12.5−4 = 8.5 m/s
| Known Values | Formula for vi | Notes |
|---|---|---|
| vf, a, t | vi = vf − at | Most common |
| vf, a, d | vi = √(vf² − 2ad) | No time needed |
| v_avg, vf | vi = 2v_avg − vf | From average |
| d, t, a | vi = d/t − ½at | From displacement |
| d, t (no a) | vi = d/t | Constant velocity |
🎯 Projectile motion: In projectile motion, initial velocity splits into two components: vi_x = v₀cosθ (horizontal, constant) and vi_y = v₀sinθ (vertical, changes with gravity). See the projectile motion section below.
Final Velocity Formula — How to Find vf
Final velocity (vf) — also written as v (UK notation) or vend — is the velocity at the END of a time interval. The variable vf appears in every kinematic equation. It can be zero (object stops), positive, or negative (reversed direction).
vf = vi + at
vf = √(vi²+2ad)
vf = 2v_avg − vi
Formula 1: vf = vi + at — primary kinematic equation for final velocity
vi=5 m/s, a=3 m/s², t=4 s → vf = 5+12 = 17 m/s
vi=5 m/s, a=3 m/s², t=4 s → vf = 5+12 = 17 m/s
Formula 2: vf = √(vi² + 2ad) — when time is unknown
vi=10 m/s, a=5 m/s², d=20 m → vf = √(100+200) = √300 = 17.32 m/s
vi=10 m/s, a=5 m/s², d=20 m → vf = √(100+200) = √300 = 17.32 m/s
Formula 3: vf = 2×v_avg − vi — from average velocity
v_avg=20 m/s, vi=10 m/s → vf = 40−10 = 30 m/s
v_avg=20 m/s, vi=10 m/s → vf = 40−10 = 30 m/s
| Known Values | Formula for vf | Notes |
|---|---|---|
| vi, a, t | vf = vi + at | Most common — use Tool 2 above |
| vi, a, d | vf = √(vi²+2ad) | No time needed — use Tool 3 above |
| v_avg, vi | vf = 2v_avg − vi | From average velocity |
How to Find Velocity Without Time
Scenario 1 — Find vf without time (know displacement):
vf² = vi² + 2ad → vf = √(vi² + 2ad)
Example 1: Ball rolls down 5 m ramp from rest, a=4 m/s²
→ vf = √(0 + 2×4×5) = √40 = 6.32 m/s (time NOT needed)
Example 2: Car brakes to rest over 30 m, deceleration=6 m/s²
→ vi = √(0 − 2×(−6)×30) = √360 = 18.97 m/s = 68.3 km/h
vf² = vi² + 2ad → vf = √(vi² + 2ad)
Example 1: Ball rolls down 5 m ramp from rest, a=4 m/s²
→ vf = √(0 + 2×4×5) = √40 = 6.32 m/s (time NOT needed)
Example 2: Car brakes to rest over 30 m, deceleration=6 m/s²
→ vi = √(0 − 2×(−6)×30) = √360 = 18.97 m/s = 68.3 km/h
Scenario 2 — Find vf without acceleration (know vi, d, t):
vf = 2×(d/t) − vi
Object travels 100 m in 8 s, starting at 5 m/s:
v_avg = 100/8 = 12.5 m/s → vf = 2×12.5 − 5 = 20 m/s
vf = 2×(d/t) − vi
Object travels 100 m in 8 s, starting at 5 m/s:
v_avg = 100/8 = 12.5 m/s → vf = 2×12.5 − 5 = 20 m/s
| What you need | What you know | Formula |
|---|---|---|
| vf (no time) | vi, a, d | vf = √(vi²+2ad) |
| vi (no time) | vf, a, d | vi = √(vf²−2ad) |
| vf (no acceleration) | vi, d, t | vf = 2(d/t) − vi |
| vi (no acceleration) | vf, d, t | vi = 2(d/t) − vf |
| vf (free fall, no time) | height h | vf = √(2gh) |
Initial Velocity vs Final Velocity — Key Differences
| Property | Initial Velocity (vi) | Final Velocity (vf) |
|---|---|---|
| Definition | Velocity at START of time interval | Velocity at END of time interval |
| Symbol (US) | vi or v₀ | vf or v |
| Symbol (UK) | u | v |
| Can equal zero? | Yes — starts from rest | Yes — comes to a stop |
| Can be negative? | Yes — moving backwards | Yes — reversed direction |
| In free fall (drop) | vi = 0 | vf = gt |
| In free fall (throw up) | vi = v₀ (upward) | vf = −vi (same magnitude, downward) |
💡 Key insight: "Initial" and "final" refer to the START and END of the specific time interval you are analyzing — not the start or end of all motion. A ball in mid-flight has an initial velocity for any time interval you choose to study.
How to Find Initial Velocity in Projectile Motion
In projectile motion the initial velocity splits into two independent components:
↔ Horizontal
vi_x = v₀ × cosθ
Constant throughout flight — no horizontal force
↕ Vertical
vi_y = v₀ × sinθ
Changes with gravity — decelerates then accelerates
Method 1 — From range R and angle θ:
v₀ = √(R×g / sin2θ)
Method 2 — From max height H and angle θ:
v₀ = √(2gH) / sinθ
Method 3 — From time of flight T and angle θ:
v₀ = g×T / (2sinθ)
Worked example: Ball lands 30 m away, launched at 40°
v₀ = √(30×9.81/sin80°) = √(294.3/0.9848) = √299 = 17.29 m/s
vi_x = 17.29×cos40° = 13.24 m/s (horizontal)
vi_y = 17.29×sin40° = 11.11 m/s (vertical)
v₀ = √(30×9.81/sin80°) = √(294.3/0.9848) = √299 = 17.29 m/s
vi_x = 17.29×cos40° = 13.24 m/s (horizontal)
vi_y = 17.29×sin40° = 11.11 m/s (vertical)
🎯 Use our Projectile Motion Calculator to solve all projectile motion problems, including range, maximum height, time of flight, and trajectory visualization.
Projectile Motion Calculator →
Worked Examples
1. Velocity from distance and time (v = d/t)
Usain Bolt: d=100 m, t=9.58 s → v = 100/9.58 = 10.44 m/s = 37.58 km/h = 23.35 mph = 20.28 knots. Check: 10.44×9.58 = 100.0 m ✓
Usain Bolt: d=100 m, t=9.58 s → v = 100/9.58 = 10.44 m/s = 37.58 km/h = 23.35 mph = 20.28 knots. Check: 10.44×9.58 = 100.0 m ✓
2. Final velocity using vf = vi + at
vi=0, a=9.81 m/s², t=3 s → vf = 0 + 9.81×3 = 29.43 m/s = 105.95 km/h = 65.84 mph. This is an object in free fall for 3 seconds.
vi=0, a=9.81 m/s², t=3 s → vf = 0 + 9.81×3 = 29.43 m/s = 105.95 km/h = 65.84 mph. This is an object in free fall for 3 seconds.
3. Final velocity without time (vf² = vi² + 2ad)
vi=20 m/s, a=−3 m/s², d=50 m → vf² = 400 + 2(−3)(50) = 400−300 = 100 → vf = √100 = 10 m/s = 36 km/h. Time: t=(10−20)/(−3) = 3.33 s
vi=20 m/s, a=−3 m/s², d=50 m → vf² = 400 + 2(−3)(50) = 400−300 = 100 → vf = √100 = 10 m/s = 36 km/h. Time: t=(10−20)/(−3) = 3.33 s
4. Average velocity (v_avg = (vi+vf)/2)
vi=0 m/s, vf=30 m/s → v_avg = (0+30)/2 = 15 m/s = 54 km/h = 33.55 mph
vi=0 m/s, vf=30 m/s → v_avg = (0+30)/2 = 15 m/s = 54 km/h = 33.55 mph
5. Velocity from kinetic energy (v = √(2KE/m))
m=1,000 kg car, KE=289,000 J → v = √(2×289,000/1,000) = √578 = 24.04 m/s = 86.5 km/h ≈ 53.8 mph
m=1,000 kg car, KE=289,000 J → v = √(2×289,000/1,000) = √578 = 24.04 m/s = 86.5 km/h ≈ 53.8 mph
6. Convert m/s to km/h and mph
v = 10 m/s → km/h: 10 × 3.6 = 36 km/h → mph: 10 × 2.237 = 22.37 mph → ft/s: 10 / 0.3048 = 32.81 ft/s → knots: 10 / 0.5144 = 19.44 knots
v = 10 m/s → km/h: 10 × 3.6 = 36 km/h → mph: 10 × 2.237 = 22.37 mph → ft/s: 10 / 0.3048 = 32.81 ft/s → knots: 10 / 0.5144 = 19.44 knots
7. Find deceleration from initial and final velocity
vi=30 m/s, vf=0 (stops), t=5 s → a = (vf−vi)/t = (0−30)/5 = −6 m/s² = −0.611 g. Negative sign confirms deceleration.
vi=30 m/s, vf=0 (stops), t=5 s → a = (vf−vi)/t = (0−30)/5 = −6 m/s² = −0.611 g. Negative sign confirms deceleration.
8. Find distance from velocity and time (d = v×t)
v=25 m/s, t=120 s → d = 25×120 = 3,000 m = 3.0 km = 1.864 miles = 9,843 ft
v=25 m/s, t=120 s → d = 25×120 = 3,000 m = 3.0 km = 1.864 miles = 9,843 ft
9. Find time from velocity and distance (t = d/v)
d=500 km, v=120 km/h → t = 500/120 = 4.167 hours = 250 minutes = 4 h 10 min
d=500 km, v=120 km/h → t = 500/120 = 4.167 hours = 250 minutes = 4 h 10 min
10. Velocity of a falling object (from height)
Drop from h=20 m: vf = √(2×9.81×20) = √392.4 = 19.81 m/s = 71.3 km/h. Time to fall: t = vf/g = 19.81/9.81 = 2.02 seconds.
Drop from h=20 m: vf = √(2×9.81×20) = √392.4 = 19.81 m/s = 71.3 km/h. Time to fall: t = vf/g = 19.81/9.81 = 2.02 seconds.
Frequently Asked Questions
What is the formula for velocity?
The basic velocity formula is v = d/t (velocity = distance ÷ time). For accelerating objects: vf = vi + at (with time) or vf² = vi² + 2ad (without time). Average velocity uses v_avg = (vi + vf)/2 for constant acceleration. Velocity from kinetic energy uses v = √(2KE/m).
How do you calculate final velocity?
Two methods: (1) When time is known: vf = vi + at. Substitute initial velocity (vi), acceleration (a), and time (t). (2) When displacement is known: vf = √(vi² + 2ad). Both give the same result for consistent inputs. The variable vf represents the velocity at the end of the time interval being studied.
What is the difference between velocity and speed?
Speed is a scalar — magnitude only (e.g. 60 km/h). Velocity is a vector — magnitude and direction (e.g. 60 km/h north). For straight-line one-direction motion they are equal in magnitude. For circular motion or return journeys they differ: average velocity on a round trip is zero while average speed is positive.
How do you find average velocity?
For constant acceleration: v_avg = (vi + vf)/2. For any motion: v_avg = total displacement / total time. Important: average velocity uses displacement (net change in position), not total distance. A round trip gives average velocity = 0 even if average speed is high.
What is the formula for average velocity?
The average velocity formula is v_avg = (vi + vf)/2 for constant acceleration, or v_avg = d_total/t_total for general motion. The formula v_avg = (vi + vf)/2 is the arithmetic mean and only works when acceleration is constant throughout the interval.
How do you calculate velocity without time?
Use the third kinematic equation: vf² = vi² + 2ad, rearranged to vf = √(vi² + 2ad). This deliberately omits time and uses displacement instead. Example: vi=20 m/s, a=−3 m/s², d=50 m → vf = √(400−300) = 10 m/s. If vi² + 2ad is negative, the object stops before reaching that distance.
What are the units of velocity?
The SI unit of velocity is metres per second (m/s). Other common units: km/h (kilometres per hour), mph (miles per hour), ft/s (feet per second), knots (nautical miles per hour). Conversion: multiply m/s by 3.6 to get km/h, by 2.237 for mph, divide by 0.3048 for ft/s, divide by 0.5144 for knots.
How do you convert m/s to km/h?
Multiply by 3.6: km/h = m/s × 3.6. Example: 10 m/s × 3.6 = 36 km/h. To reverse: divide km/h by 3.6 to get m/s. The factor 3.6 comes from 1,000 m/km ÷ 3,600 s/hour = 1/3.6 m/s per km/h.
What is the kinematic equation for final velocity?
There are two: vf = vi + at (when time is known) and vf² = vi² + 2ad (when displacement is known instead of time). In UK notation: v = u + at and v² = u² + 2as. Both are part of the three SUVAT equations taught in physics. The first equation is linear in time; the second eliminates time entirely.
What does vf mean in physics?
vf means "final velocity" — the velocity of an object at the end of a specific time interval. The subscript 'f' stands for 'final'. It is also written as v (UK notation) or vend. vf can be zero (object stops), positive (moving forward), or negative (moving backward). The corresponding starting velocity is vi (initial velocity), also written u in UK notation.