Velocity Calculator
Solve velocity using Average, Instantaneous (from x(t)), Constant Acceleration, or 2D Vectors. Includes unit conversions, direction/sign, quick picks, and optional step-by-step.
Background
Velocity is speed with direction. In 1D, direction is often a sign (+/−). In 2D, direction is an angle and the velocity can be represented by components vx, vy.
How this calculator works
- Average velocity: v = Δx/Δt
- Instantaneous velocity: v(t) = dx/dt (computed numerically)
- Constant acceleration: v = v₀ + a·t or v² = v₀² + 2aΔx
- 2D velocity: |v| = √(vₓ² + vᵧ²), θ = atan2(vᵧ, vₓ)
Units: This tool converts everything internally to SI (meters, seconds) to avoid unit mistakes.
Formula & Equation Used
Average velocity: v = Δx/Δt
Instantaneous velocity: v(t) = dx/dt
Numerical derivative (central difference): v(t₀) ≈ [x(t₀+h) − x(t₀−h)]/(2h)
Constant acceleration: v = v₀ + a·t
Constant acceleration (time-free): v² = v₀² + 2aΔx
2D magnitude: |v| = √(vₓ² + vᵧ²)
2D direction: θ = atan2(vᵧ, vₓ) (angle from +x axis)
Polar → components: vₓ = |v|cosθ, vᵧ = |v|sinθ
Example Problems & Step-by-Step Solutions
Example 1 — Average velocity (v = Δx/Δt)
Given Δx = 120 m and Δt = 6 s, find v.
- Use: v = Δx/Δt
- Substitute: v = 120/6
- Result: v = 20 m/s (≈ 72 km/h)
Example 2 — Instantaneous velocity from x(t)
Given x(t) = 3t² − 2t + 1 (meters) at t₀ = 2 s, find v(t₀).
- Conceptually, v(t) = dx/dt.
- Derivative (for reference): v(t) = 6t − 2.
- Evaluate: v(2) = 6·2 − 2 = 10
- Result: 10 m/s
Note: Your calculator computes this using a central-difference approximation (very close to the exact value for smooth functions).
Example 3 — Constant acceleration (v = v₀ + a·t)
Given v₀ = 4 m/s, a = −1.5 m/s², and t = 6 s, find v.
- Use: v = v₀ + a·t
- Substitute: v = 4 + (−1.5)·6
- Compute: v = 4 − 9 = −5
- Result: −5 m/s (negative means opposite direction)
Example 4 — 2D vector velocity from components
Given vₓ = 3 m/s and vᵧ = 4 m/s, find |v| and θ.
- Magnitude: |v| = √(3² + 4²) = √25 = 5 → 5 m/s
- Angle: θ = atan2(4, 3) ≈ 53.13°
Frequently Asked Questions
Q: What’s the difference between speed and velocity?
Speed is how fast you move (a magnitude). Velocity includes direction. In 1D that direction is often shown by a sign (+/−); in 2D it’s an angle (or x/y components).
Q: Why can average velocity be 0 even if I moved?
Average velocity uses displacement (net change in position), not total distance. If you go out and come back, your displacement is 0, so v = Δx/Δt = 0.
Q: Why does the calculator require Δt > 0?
Because v = Δx/Δt would be undefined if Δt = 0, and negative time usually indicates a units/input mistake.
Q: In 2D mode, should I use degrees or radians?
Use degrees for typical physics problems unless the problem explicitly gives radians. The calculator will convert internally and still show the angle relative to the +x axis.
Q: Why does the v² equation mention ±v?
Because v² = v₀² + 2aΔx gives a magnitude squared. When you take the square root, both +v and −v are mathematically possible—your scenario (chosen positive direction) determines the correct sign.
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