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Physics Applied Mechanics

Classical Mechanics · Applied to Human Movement

The Physics of Sports

Every jump shot, javelin throw, figure-skating spin, and tackle is a physics problem the body solves in real time. Biomechanics is where kinematics, forces, torque, and energy stop being abstract equations and start explaining why a bent knee absorbs a landing, why skaters pull in their arms to spin faster, and why 45° isn't always the best angle to throw at.

Kinematics · Forces · Torque · Energy 2 Simulations Grades 9–12
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You can know the name of a bird in all the languages of the world, but when you're finished, you'll know absolutely nothing whatever about the bird. So let's look at the bird and see what it's doing — that's what counts.

— Richard Feynman

Idea 1 · Center of Mass & Stability

Balance Is a Physics Problem

Every object — including a human body — has a center of mass: the single point where all its weight can be thought of as acting. An athlete is stable as long as their center of mass stays above their base of support (the area under and between the feet, hands, or skates touching the ground).

A wide stance, a low crouch, and a wide base of support all lower and enlarge that stability zone — which is exactly why a wrestler crouches, a sprinter drives low out of the blocks, and a football lineman keeps their center of gravity close to the ground.

The Fosbury Flop

High jumpers arch their backs over the bar so violently that their center of mass actually passes underneath the bar while their body passes over it. The athlete clears a bar their own center of mass never crosses — a trick of geometry, not extra jump height.

Why Athletes Widen Their Stance

A narrow base of support means a small torque can tip the center of mass past the edge of that base — and once gravity's torque acts outside the base, balance is lost and rotation (a fall) begins. Widening the stance buys more angular room before that happens.

  • Gymnasts stick a landing by absorbing momentum through bent knees while keeping their center of mass over their feet.
  • Surfers & skateboarders constantly shift their center of mass to stay above a moving base of support.
  • Sumo wrestlers use an extremely wide, low stance to maximize the torque an opponent must generate to topple them.

Idea 2 · Projectile Motion in Sport

Every Throw, Jump, and Shot Is a Parabola

A basketball, a javelin, a long jumper's body in flight — once they leave contact with the ground or the athlete's hand, only gravity acts on them (ignoring air resistance). Horizontal velocity stays constant; vertical velocity changes at g = 9.8 m/s². The result is always a parabola.

Accounting for the height the object is released from (h₀) above the landing point:

y(t) = h₀ + v₀·sin(θ)·t − ½g·t²
x(t) = v₀·cos(θ)·t

Why 45° Isn't Always Best

For a launch and landing at the same height, 45° maximizes range. But most sports release the object above the landing point — a basketball leaves the hand near the shoulder or above the head; a shot put leaves the hand near the shoulder, landing at ground level. That extra release height means the optimal angle drops below 45°.

Real Release Angles

Elite shot putters release near 38–42°, not 45° — the extra height and the speed they can generate at lower angles wins out. Basketball free throws are typically shot near 45–55° to arc over the front rim and drop steeply into the basket.

🏀 Sport Launch Lab Release height changes the optimal angle — try a preset, then hunt for the max-range angle yourself
Range Max Height Air Time

Idea 3 · Torque & Rotational Motion

Twisting Forces: Bats, Swings, and Joints

A force applied off-center doesn't just push — it rotates. That rotational effect is torque: the product of the force, the distance from the pivot (the lever arm), and the angle between them.

τ = F · r · sin(θ)

Every human joint is a lever. A bicep curling a dumbbell, a pitcher's shoulder whipping a baseball, a golfer's hips rotating before the arms — each is torque generated at a pivot and transmitted outward.

The "Sweet Spot" on a Bat

A baseball or cricket bat has a center of percussion — the point where an impact produces the least jarring reaction force back at the hands (the grip acts like a second pivot). Hit there, and almost all the bat's rotational energy transfers to the ball instead of stinging your palms.

Kinetic Chain

A golf swing or pitch isn't one motion — it's a sequence: legs and hips rotate first, then torso, then shoulder, then arm, then wrist, each segment adding speed to the next like a whip cracking. Coaches call this sequencing the kinetic chain, and it's why swing timing matters more than raw muscle.

Idea 4 · Conservation of Angular Momentum

Why Skaters Spin Faster With Arms In

Just as linear momentum (p = mv) is conserved with no outside force, angular momentum (L = Iω) is conserved with no outside torque. Here I is moment of inertia — how spread out an object's mass is from its spin axis — and ω (omega) is angular velocity, how fast it spins.

L = I · ω   (constant when no external torque acts)

Pull mass closer to the spin axis and I shrinks. Since L must stay constant, ω has to grow to compensate — the spinner speeds up automatically.

I
Moment of Inertia
Mass × (distance from axis)². Arms out = large I. Arms tucked in = small I.
ω
Angular Velocity
Rotation rate, in radians per second. Grows as I shrinks to keep L constant.
Same Trick, Different Sports

Figure skaters pulling in their arms, divers tucking into a somersault, and even a falling cat twisting to land on its feet all exploit the same conservation law. No muscle makes them spin faster — repositioning mass around the spin axis does.

⛸️ Spin Lab: Conservation of Angular Momentum Start the spin, then drag the arm slider in and out — watch ω change while L stays fixed
I (rel.) ω (rad/s) L (rel., constant)

Idea 5 · Impulse & Impact

Why Bending Your Knees Saves Your Bones

Impulse is force acting over time, and it always equals the change in momentum — this is Newton's Second Law rearranged for collisions and impacts.

F · Δt = Δp = mΔv

A given athlete landing from a given jump has a fixed Δp — their momentum has to go to zero either way. The only variable is Δt, the time the impact takes. Stretch that time out, and the average force drops proportionally.

Extending Time to Cut Force

  • Bending your knees when landing spreads the stop over a longer time than a stiff-legged landing — dramatically lowering peak force on joints.
  • Boxing gloves and padding extend impact time so the same punch delivers less peak force to bone and brain.
  • Catching a ball with "give" — pulling your hands back as it arrives — softens the catch the same way.
The Flip Side: Follow-Through

In a golf swing or a punch, athletes want more impulse delivered to the ball or target, not less. A longer follow-through keeps the club or fist in contact (or accelerating) longer, increasing Δt on the giving end and maximizing the momentum transferred.

Idea 6 · Energy Transfer in Sport

Springs, Tendons, and Stored Energy

Muscles aren't the only thing powering explosive movement — tendons and equipment store and return elastic potential energy like a spring.

KE = ½mv²    PE_elastic = ½kx²

A pole vaulter's kinetic energy from the sprint bends the pole, storing elastic energy, which then converts to gravitational potential energy as the pole straightens and launches the vaulter upward — a two-stage energy relay.

Built-In Springs

  • Achilles tendon stores and returns up to 35% of the energy in each running stride — a big reason running is far more efficient than walking at speed.
  • Trampolines & diving boards store an athlete's kinetic energy as elastic energy, then return it to launch them higher than their own muscles could.
  • Running shoes & track surfaces are engineered to maximize elastic energy return with each footstrike.
The Big Picture

Biomechanics shows that world-class performance isn't just bigger muscles — it's better physics: lower center of mass, smarter release angles, sequenced torque, conserved angular momentum, longer impulse times, and elastic energy return, all working together.