SolveMyPhysics

Projectile Motion Calculator

Enter a launch speed and angle to get range, maximum height, time of flight and both velocity components, in whichever units you have.

range = v₀²·sin(2θ) ÷ g    height = (v₀·sin θ)² ÷ 2g    time = 2·v₀·sin θ ÷ g

Range

40.79 m

Max height

10.2 m

Time of flight

2.884 s

Velocity components

horizontal vₓ = 14.14 m/s  ·  vertical v_y = 14.14 m/s

Gravity is taken as the exact standard value g = 9.80665 m/s². Air resistance is ignored, which is the standard textbook treatment — a real thrown or fired object falls short of these figures, and the gap widens the faster and lighter it is.

Two Motions at Once

A projectile does two independent things simultaneously. Horizontally it moves at a constant speed, because nothing pushes it sideways once released. Vertically it is in free fall, decelerating on the way up and accelerating on the way down at g. Splitting the launch velocity into those two components is the whole technique.

The horizontal component is v₀·cos θ and the vertical is v₀·sin θ. From those: time of flight is 2·v₀·sin θ ÷ g, maximum height is (v₀·sin θ)² ÷ 2g, and range is v₀²·sin(2θ) ÷ g.

Worked example — thrown at 20 m/s

Launched at 20 m/s at 45°, the optimum angle:

Range = 40.79 m, peak height 10.2 m, airborne for 2.884 s.

The symmetry, demonstrated. At 30° the range is 35.32 m; at 60° it is 35.32 m — the same distance to within rounding. But the 60° throw peaks at 15.3 m against just 5.099 m for the 30° throw, and hangs in the air 1.7 times as long.

A second example at a very different speed. 100 mph is 44.7 m/s. Launched at 45°, that reaches 203.8 m — because range scales with the square of launch speed, roughly 2.2× the speed gives about 5× the distance.

Range and Height by Launch Angle

All rows computed at a fixed 20 m/s launch speed, so only the angle varies.

AngleRangeMax heightFlight time
15°20.39 m1.366 m1.06 s
30°35.32 m5.099 m2.04 s
45°40.79 m10.2 m2.88 s
60°35.32 m15.3 m3.53 s
75°20.39 m19.03 m3.94 s
90°0 m20.39 m4.08 s

Note the 90° row: launched straight up, the range is zero and the projectile lands back on the launcher. Maximum height peaks there while range peaks at 45°, which is the whole trade-off in one table.

Keep going

  • The vertical half of a projectile's path is ordinary free fall; the free fall calculator isolates that component on its own. Free Fall Calculator
  • A projectile's launch speed splits into horizontal and vertical velocity components; the velocity calculator works with those components directly. Velocity Calculator

Frequently Asked Questions

What launch angle gives the maximum range?

45 degrees, when landing at the same height you launched from. The range formula v₀²·sin(2θ) ÷ g peaks where sin(2θ) = 1, which is 2θ = 90°, so θ = 45°. Launching from above the landing height shifts the optimum slightly below 45°.

Why do 30 and 60 degrees give the same range?

Because sin(2θ) is symmetric about 90°: sin(60°) and sin(120°) are equal, so 30° and 60° produce identical ranges. The paths differ completely — the 60° shot goes much higher and stays airborne longer — but they land in the same place.

Does the projectile's mass matter?

Not in a vacuum. Mass appears nowhere in the range, height or flight-time formulas, because gravity accelerates every object identically. In real air it matters a great deal, since drag depends on mass, size and shape — a golf ball and a balloon launched identically behave nothing alike.

How high does a projectile go?

h = (v₀·sin θ)² ÷ 2g. Only the vertical component of the launch velocity contributes, which is why a flat line drive barely rises while a lofted shot at the same speed climbs steeply.

Is air resistance included?

No. These are the standard vacuum equations. Real projectiles fall well short of these ranges — a struck golf ball or fired bullet can lose a large fraction of its vacuum range to drag — so treat the output as an upper bound and a teaching tool rather than a ballistics prediction.

Related Calculators

View all calculators →