Projectile Motion — Scivra Labs

TRY THIS FIRST

Which launch angle throws the ball the farthest?

Most people guess "steeper is farther." Set the angle to 20°, then 45°, then 75° with the same speed and compare the range readout.

→ You'll find 45° wins — and that 30° and 60° land at the exact same spot.

CORE FORMULAS

Horizontal position

Vertical position

Range

Max height

Total mechanical energy (conserved without drag)

THEORY

Projectile motion is the motion of an object thrown into the air, subject only to gravity. The horizontal and vertical components of motion are independent. The trajectory forms a parabola.

HOW TO USE

Set the initial velocity and launch angle. Press play to fire the projectile. Observe the trajectory trace and compare range at different angles. Add air resistance to watch mechanical energy dissipate; increase gravity to shrink the range.

LEARNING CARDS

Independence of Axes

Horizontal velocity is constant (no horizontal force without drag), while vertical velocity changes at −g every second. The two axes never talk to each other — that is why the shadow on the ground moves at a steady pace while the height rises and falls.

Why 45° Is Optimal

Range depends on sin(2θ), which peaks at 2θ = 90°, i.e. θ = 45°. Because sin(2θ) is symmetric about 45°, complementary angles like 30° and 60° produce identical ranges — one flat and fast, one high and slow.

Energy Along the Arc

Kinetic energy is greatest at launch and landing; potential energy peaks at the apex. Without air resistance their sum stays constant — a perfectly elastic bounce here conserves it forever so you can watch KE and PE trade back and forth.

Gravity Sets the Scale

Range is inversely proportional to g. On Mars (g = 3.7 m/s²) the same throw travels about 2.6× farther than on Earth (9.8 m/s²), because R ∝ 1/g. Lower gravity means longer hang time and a flatter-looking arc.

WHY IT MATTERS

Ballistics and sports — the launch angle that maximizes distance for a javelin, basketball, or cannonball
Spacecraft landing — projectile physics scales from a thrown ball to a suborbital hop on another world
Engineering — fountains, sprinklers, and water jets are all designed around the parabolic range equation

🔗 AP Physics 1 alignment: 3.A.1 (kinematics in two dimensions), 3.E.1 (work–energy). NGSS: HS-PS2-1. GCSE: P5.6.

ENERGY

KE0.000 J

PE0.000 J

Total0.000 J

LIVE DATA

R (theory)	—	m
H (theory)	—	m
t flight	—	s
x	—	m
y	—	m
speed	—	m/s
E_tot	—	J

QUIZ

Speed:

Preset:

Velocity v₀: 30 m/s

Angle θ: 45 °

Gravity g: 9.8 m/s²

Mass m: 1.0 kg

Air resistance: 0.00 /s