How do you calculate the range of a projectile?

R = v² × sin(2θ) ÷ g for a launch on level ground. 30 m/s at 40°: 900 × sin 80° ÷ 9.81 = 900 × 0.985 ÷ 9.81 = 90.3 m. Enter speed and angle and the page shows the substitution.

What angle gives the maximum range?

45°, when there is no air resistance — sin 2θ is largest there. Any pair of angles adding to 90° gives the same range, so 20° and 70° match. With air resistance the best angle is lower, around 38° for a cannonball in OpenStax's example, and lower still for light balls.

How do you find the maximum height and time of flight?

Height: v² sin²θ ÷ 2g — 20 m/s at 45° peaks at 10.2 m. Time: 2v sinθ ÷ g — 2.88 s for the same throw. Both come from the vertical component v sinθ alone; the horizontal speed does not affect them.

Does a heavier projectile go further?

Not in a vacuum — mass cancels out of every projectile formula, so a cannonball and a marble at the same speed and angle land together. In air the heavier one goes further because drag slows it proportionally less; that is real, but it is air, not gravity.

Projectile Motion Calculator (Range, Max Height, Time of Flight)

Range, maximum height and time of flight of a projectile from launch speed and angle, no air resistance, working shown.

Inputs

Launch speed(m/s)

0.001 m/s1,000,000 m/s

Launch angle(°)

0 °90 °

Gravity(m/s²)

0.1 m/s²100 m/s²

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Horizontal range

40.7886 m

Working: R = v² sin(2θ) ÷ g = 20² × sin(90°) ÷ 9.80665 = 400.00 × 1 ÷ 9.80665 = 40.7886 m
Max height: v² sin²θ ÷ 2g = 10.1972 m
Time of flight: 2v sinθ ÷ g = 2.8842 s
Velocity components at launch: horizontal 14.1421 m/s, vertical 14.1421 m/s
Range at 45° with this speed (the maximum): 40.7886 m

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What each Projectile Motion input means

Launch speed: The launch speed in m/s — how fast it leaves the hand, bat or barrel. km/h ÷ 3.6 converts.
Launch angle: The launch angle above the horizontal, from 0° (flat) to 90° (straight up).
Gravity: Gravity, 9.80665 m/s² on Earth by default; 1.62 on the Moon, 3.72 on Mars.

Projectile Motion formula

range R = v² sin(2θ) ÷ g · max height H = v² sin²θ ÷ 2g · time of flight T = 2v sinθ ÷ g · (level ground, no air)

R = \frac{v_0^{2} \sin 2\theta_0}{g}, \quad H = \frac{v_0^{2} \sin^{2}\theta_0}{2g}, \quad T = \frac{2 v_0 \sin\theta_0}{g}

A projectile launched at speed v and angle θ moves horizontally at a steady v cos θ while gravity acts only on the vertical part, v sin θ, which is why the two directions are worked separately. The vertical motion decides the time: the ball rises for v sinθ ÷ g seconds and falls for as long again, so the flight lasts 2v sinθ ÷ g and the peak is v² sin²θ ÷ 2g high. The horizontal motion decides the distance: constant speed times that time, which simplifies to v² sin 2θ ÷ g. Because sin 2θ peaks at 90°, the range is greatest at a 45° launch and the same at any two angles that add to 90° — 30° and 60° land in the same place. All of this assumes level ground and no air; the page shows the substitution, the velocity components and what 45° would have given.

Projectile Motion worked example

20 m/s at 45°: R = 20² × sin 90° ÷ 9.807 = 400 × 1 ÷ 9.807 = 40.8 m, the maximum this speed can reach. The peak is 400 × sin²45° ÷ (2 × 9.807) = 10.2 m and the flight lasts 2 × 20 × 0.707 ÷ 9.807 = 2.88 s. At 30° the same throw goes 400 × sin 60° ÷ 9.807 = 35.3 m and peaks at only 5.1 m; at 60° it also goes 35.3 m but peaks at 15.3 m.

More Projectile Motion examples

A ball struck at 30 m/s and 40°: R = 30² × sin 80° ÷ 9.80665 = 900 × 0.9848 ÷ 9.80665 = 90.3802 m. It peaks at 19 m and stays up for 3.93 s; at 45° the same speed would reach 91.8 m.

Where the Projectile Motion calculator can give the wrong answer

No air resistance is a large assumption for anything light or fast. A golf ball, a football or a bullet goes much less far than the formula says and its best angle drops well below 45° — OpenStax quotes about 38° with air.
Level ground only. Launching from a height, or landing on a hill, changes the range; the general case needs the full trajectory equation rather than R = v² sin 2θ ÷ g.
The launch speed is the speed as it leaves the hand, bat or barrel, not the speed of the arm or the muzzle energy; it is the hardest number to know.
Angles are measured from the horizontal. 0° is a flat throw with zero range on level ground (it lands instantly) and 90° goes straight up and comes straight down.
Gravity is 9.80665 m/s² by default; change it for the Moon (1.62) and the same throw goes six times as far.

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Projectile Motion reference table

A 20 m/s launch at different angles (level ground, no air)
Angle	Range	Maximum height	Time of flight
15°	20.4 m	1.4 m	1.06 s
30°	35.3 m	5.1 m	2.04 s
45°	40.8 m	10.2 m	2.88 s
60°	35.3 m	15.3 m	3.53 s
75°	20.4 m	19 m	3.94 s

Source: Worked out with this page's formulas, g = 9.80665 m/s² (OpenStax College Physics 2e, §3.4)

Projectile Motion FAQ

: R = v² × sin(2θ) ÷ g for a launch on level ground. 30 m/s at 40°: 900 × sin 80° ÷ 9.81 = 900 × 0.985 ÷ 9.81 = 90.3 m. Enter speed and angle and the page shows the substitution.
: 45°, when there is no air resistance — sin 2θ is largest there. Any pair of angles adding to 90° gives the same range, so 20° and 70° match. With air resistance the best angle is lower, around 38° for a cannonball in OpenStax's example, and lower still for light balls.
: Height: v² sin²θ ÷ 2g — 20 m/s at 45° peaks at 10.2 m. Time: 2v sinθ ÷ g — 2.88 s for the same throw. Both come from the vertical component v sinθ alone; the horizontal speed does not affect them.
: Not in a vacuum — mass cancels out of every projectile formula, so a cannonball and a marble at the same speed and angle land together. In air the heavier one goes further because drag slows it proportionally less; that is real, but it is air, not gravity.