Resultant Velocity Calculator
Calculate vector addition of velocities with magnitude and direction
Calculate Resultant Velocity
Velocity Vector 1
Velocity Vector 2
Resultant Velocity
Vector Analysis
Vector Sum Verification
Vector Addition Formulas
Component Sum: vₓ = Σ(vᵢ cos θᵢ), vᵧ = Σ(vᵢ sin θᵢ)
Magnitude: |v⃗| = √(vₓ² + vᵧ²)
Direction: θ = arctan(vᵧ/vₓ)
Where vᵢ and θᵢ are the magnitude and angle of each velocity vector
Vector Physics Insights
Example: Boat Crossing River
Problem Setup
Scenario: Boat crossing a river with current
Boat velocity: 15 km/h straight across (0°)
River current: 7 km/h downstream (90°)
Question: What is the boat's resultant velocity?
Solution Steps
1. Calculate x-components: vₓ = 15×cos(0°) + 7×cos(90°) = 15 + 0 = 15 km/h
2. Calculate y-components: vᵧ = 15×sin(0°) + 7×sin(90°) = 0 + 7 = 7 km/h
3. Find magnitude: |v⃗| = √(15² + 7²) = √(225 + 49) = √274 ≈ 16.55 km/h
4. Find direction: θ = arctan(7/15) = arctan(0.467) ≈ 25°
Result: Boat moves at 16.55 km/h at 25° from its intended direction
Real-World Examples
Boat Crossing River
Boat velocity + river current
Airplane with Wind
Aircraft velocity + wind velocity
Walking on Moving Train
Train velocity + passenger velocity
Projectile Motion Components
Initial velocity components
Vector Physics Concepts
Vector Addition
Tip-to-tail method or component addition
Components
Break vectors into x and y parts
Resultant
Net effect of all vectors combined
Direction
Angle from positive x-axis
Essential Vector Formulas
Vector Components
vₓ = v cos θ
vᵧ = v sin θ
Resultant Magnitude
|v⃗| = √(vₓ² + vᵧ²)
Pythagorean theorem
Resultant Direction
θ = arctan(vᵧ/vₓ)
Angle from x-axis
Vector Addition
v⃗ᵣₑₛ = v⃗₁ + v⃗₂ + ... + v⃗ₙ
Sum all component vectors
Understanding Resultant Velocity
What is Resultant Velocity?
Resultant velocity is the vector sum of all individual velocity vectors acting on an object. It represents the net velocity that combines all the separate motions into a single equivalent velocity.
Vector Addition Principles
Velocities are vectors with both magnitude and direction. When adding vectors, we must consider both components. The resultant is found by adding all x-components and all y-components separately.
Component Method
Each velocity vector is broken into horizontal (x) and vertical (y) components using trigonometry. The resultant components are summed, then combined to find the final magnitude and direction.
Real-World Applications
Navigation (ships, aircraft), relative motion problems, projectile motion analysis, and fluid dynamics all rely on vector addition of velocities for accurate calculations.
Navigation Applications
- • Aircraft velocity with wind
- • Ship velocity with current
- • GPS navigation systems
- • Maritime route planning
Physics Applications
- • Relative motion analysis
- • Projectile motion problems
- • Collision calculations
- • Fluid dynamics studies
Engineering Uses
- • Vehicle motion planning
- • Robotics path calculation
- • Aerospace trajectory design
- • Game physics engines