What formulas does this calculator use?

For launch and landing at the same height with no air resistance: vx = v0*cos(theta), vy = v0*sin(theta), time of flight T = 2*vy/g, max height H = vy^2/(2g), range R = v0^2*sin(2*theta)/g. The calculator inverts these to solve for any two unknowns when the other three are constrained.

Which launch angle gives the greatest range?

45 degrees gives the maximum horizontal range on level ground with no drag, because sin(2*theta) reaches its peak of 1 at 2*theta = 90 degrees. Complementary angle pairs such as 30 and 60 degrees produce equal range because sin(60) = sin(120).

Why do 30 and 60 degrees give the same range?

Range is proportional to sin(2*theta). Because sin(60 degrees) = sin(120 degrees), and 2*30 = 60 while 2*60 = 120, both angles share the same sine value and therefore the same range. The 60-degree shot travels higher and stays in the air longer, but lands at the same spot.

Is air resistance included?

No. This is the idealised vacuum model taught in introductory physics. Real trajectories shorten at high speeds because drag decelerates the projectile — the effect is negligible at low speeds (a tennis ball gently thrown) but substantial at high speeds (a golf ball or bullet).

How does solve-for-any-variable work?

Each pair of known quantities uniquely determines the remaining three through algebraic inversion of the standard kinematic equations. For example, given range R and flight time T, horizontal velocity is simply R/T and vertical initial velocity is g*T/2, from which angle and speed follow directly.

How do I model the Moon or Mars?

Change gravity to 1.62 m/s^2 for the Moon or 3.72 m/s^2 for Mars using the preset buttons or by typing the value. Lower gravity means longer flight, greater height and more range for the same launch. At 1.62 m/s^2 a 20 m/s, 45-degree throw travels about 250 m instead of 40 m on Earth.

Projectile Motion Calculator

A projectile motion calculator that handles every combination of known variables: give it any two of launch speed, launch angle, range, maximum height or flight time and it derives all five quantities plus the horizontal and vertical velocity components. The result panel shows a step-by-step derivation and an interactive parabolic trajectory chart so you can see the full arc, hover for coordinates, and compare how angle changes reshape the curve.

The tool covers the idealised model taught in introductory physics and engineering courses: constant horizontal velocity, uniform gravitational acceleration, level launch and landing, no air resistance. It is suited for physics homework, quick sanity checks on launch angles, sports science estimates, game-physics prototyping and “what if we were on the Moon?” demonstrations.

How it works

The launch velocity v₀ is decomposed into two independent components:

vₓ = v₀ · cos θ      (horizontal — constant throughout the flight)
v_y = v₀ · sin θ     (vertical — reduced by gravity at rate g)

From those components, three classical results follow for a trajectory that starts and ends at the same height:

Time of flight   T = 2 · v_y / g
Maximum height   H = v_y² / (2g)
Range            R = v₀² · sin(2θ) / g

Because each formula is algebraically invertible, the calculator can back-calculate in any direction. Given range R and flight time T, for instance, it recovers vₓ = R/T and v_y = gT/2, then reconstructs speed and angle. All ten distinct two-known-quantity pairs are supported.

Worked example — 20 m/s at 45°, Earth gravity

A ball is thrown at 20 m/s at 45° on Earth (g = 9.81 m/s²):

vₓ = 20 · cos 45° = 14.14 m/s
v_y = 20 · sin 45° = 14.14 m/s
T = 2 × 14.14 / 9.81 = 2.88 s
H = 14.14² / (2 × 9.81) = 10.19 m
R = 20² × sin 90° / 9.81 = 40.77 m

Angle	Range	Max height	Flight time
30°	35.31 m	5.10 m	1.44 s
45°	40.77 m	10.19 m	2.88 s
60°	35.31 m	15.29 m	3.53 s
75°	20.38 m	18.64 m	3.93 s

Notice 30° and 60° share the same 35.31 m range — a classic result that follows from sin(60°) = sin(120°).

On the Moon (g = 1.62 m/s²)

The same 20 m/s throw at 45° on the lunar surface:

T = 2 × 14.14 / 1.62 = 17.46 s
H = 14.14² / (2 × 1.62) = 61.72 m
R = 20² × 1 / 1.62 = 246.9 m

Six times the range and six times the height — a vivid illustration of why low gravity makes sport so interesting to imagine off-Earth.

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