Essential Biomechanics Formulas and Human Motion Principles

Posted on Nov 1, 2025 in Physics

Core Biomechanics Formulas for Human Motion Analysis

1. Kinematics (Motion Without Forces)

Kinematics describes motion without considering the forces that cause it.

Key Linear Formulas:

• Displacement (Δx):

  Δx = x₂ − x₁

  Example: If a sprinter moves from 2 m to 8 m → Δx = 6 m

• Velocity (v):

  v = Δx / Δt

  Example: 6 m in 2 s → v = 6 / 2 = 3 m/s

• Acceleration (a):

  a = Δv / Δt

  Example: Speed changes from 2 m/s to 6 m/s in 2 s → a = (6−2)/2 = 2 m/s²

Angular Motion Formulas:

• Angular Displacement (θ):

  θ (rad) = arc length / radius

• Angular Velocity (ω):

  ω = Δθ / Δt

• Angular Acceleration (α):

  α = Δω / Δt

  Example: A joint rotates 90° (π/2 rad) in 0.5 s → ω = π/2 ÷ 0.5 ≈ 3.14 rad/s

2. Kinetics (Forces Causing Motion)

Kinetics analyzes the forces that produce, arrest, or modify motion.

• Newton’s Second Law (Force):

  F = ma

  Example: 60 kg athlete accelerating at 2 m/s² → F = 60 × 2 = 120 N

• Torque (τ):

  τ = F × r × sin(θ)

  Example: 50 N force applied 0.4 m from joint at 90° angle → τ = 50 × 0.4 × sin(90°) = 20 Nm

• Impulse and Momentum (Δp):

  Impulse = F × t = Δp (Change in Momentum)

  Example: A 70 kg runner changes velocity from 0 to 5 m/s in 0.2 s → Δp = 70 × 5 = 350 Ns; F = 350 / 0.2 = 1750 N

• Work (W):

  W = F × d × cos(θ)

  Example: 100 N force moves object 2 m forward → W = 100 × 2 = 200 J

• Power (P):

  P = W / t

  Example: 200 J of work done in 2 s → P = 200 / 2 = 100 W

3. Mechanical Advantage and Levers

Mechanical Advantage (MA) determines the efficiency of a lever system.

MA = Effort Arm / Resistance Arm

• MA > 1: Provides a force advantage (less effort needed).
• MA < 1: Provides a speed and Range of Motion (ROM) advantage.

Types of Levers:

4. Center of Mass (COM) and Stability

• Center of Mass (COM): The average location of the body’s mass.
• Stability Principle: Lowering the COM and widening the base of support increases stability.
• Example: Squatting lowers the COM, resulting in increased stability.

5. Gait Cycle Analysis

The gait cycle describes the sequence of events that occur between successive heel contacts of the same foot.

• Stride: Equivalent to 2 steps (heel contact of one foot to the next heel contact of the same foot).
• Cadence: Measured in steps per minute.

6. Free Body Diagrams (FBDs)

FBDs are essential for visualizing and analyzing external forces acting on a body segment.

Steps for Drawing an FBD:

1. Identify the object or system being analyzed.
2. Draw all external forces acting on the object (e.g., gravity, normal force, friction, muscle force, joint reaction force).
3. Use the equations of motion (ΣF = ma or Στ = Iα) for quantitative analysis.

7. Inverse Dynamics

Inverse dynamics is a method used to calculate internal joint forces and torques based on known motion and external forces.

• Process: Use known motion data (acceleration, joint angles) and measured external forces (e.g., Ground Reaction Force, GRF) to compute internal torques.
• Example: Knowing GRF + limb mass + joint angle allows calculation of the knee joint torque.

8. Tools in Biomechanical Analysis

9. Work, Energy, and Power

Energy is the capacity to do work.

• Kinetic Energy (KE): Energy due to motion.

  KE = ½mv²

  Example: 70 kg mass moving at 5 m/s → KE = 0.5 × 70 × 5² = 875 J

• Potential Energy (PE): Energy due to position or height.

  PE = mgh

  Example: 70 kg mass at 2 m height → PE = 70 × 9.81 × 2 ≈ 1373.4 J

• Power (P): The rate at which work is done.

  P = Work / Time

  Alternative Formula: P = F × v (Force times Velocity)

10. Key Biomechanical Units

Trigonometry Fundamentals (SOHCAHTOA)

SOHCAHTOA is a mnemonic device used to remember the definitions of the three basic trigonometric ratios in a right-angled triangle:

• SOH: Sine (θ) = Opposite / Hypotenuse
• CAH: Cosine (θ) = Adjacent / Hypotenuse
• TOA: Tangent (θ) = Opposite / Adjacent

Right-Angled Triangle Definitions:

• Opposite: The side opposite the angle θ.
• Adjacent: The side next to angle θ (but not the hypotenuse).
• Hypotenuse: The side opposite the right angle; always the longest side.

Example Problem: Calculating Sides

Given: A right-angled triangle with an angle θ = 30°, and the length of the hypotenuse is 10 units.

Find: The lengths of the opposite and adjacent sides.

1. Opposite side calculation:
   • sin(θ) = Opposite / Hypotenuse
   • sin(30°) = Opposite / 10
   • 0.5 = Opposite / 10
   • Opposite = 0.5 × 10 = 5 units
2. Adjacent side calculation:
   • cos(θ) = Adjacent / Hypotenuse
   • cos(30°) = Adjacent / 10
   • 0.866 = Adjacent / 10
   • Adjacent = 0.866 × 10 ≈ 8.66 units

Practice Problems: Applying SOHCAHTOA

1. Problem 1: Finding the Hypotenuse

   In a right-angled triangle, the angle θ = 45°, and the adjacent side is 7 units. Find the length of the hypotenuse.

   Solution:

   • cos(θ) = Adjacent / Hypotenuse
   • cos(45°) = 7 / Hypotenuse
   • 0.7071 = 7 / Hypotenuse
   • Hypotenuse = 7 / 0.7071 ≈ 9.9 units

2. Problem 2: Finding the Angle

   A right-angled triangle has an opposite side of 4 units and an adjacent side of 3 units. Find angle θ.

   Solution:

   • tan(θ) = Opposite / Adjacent
   • tan(θ) = 4 / 3
   • θ = arctan(4 / 3) ≈ 53.13°