Projectile Range Calculator

Definition of projectile range

Range Z is the horizontal distance between the place where the object leaves a level and the place where it returns to that level. The launch speed v₀ is inclined at an angle α to the horizontal. In the classroom model the horizontal acceleration is zero and the vertical acceleration is constant and equal to −g. The path is then a parabola, and the range on one level has the compact form Z = v₀² sin(2α)/g.

The peak height H_max = v₀² sin²(α)/(2g) describes the same flight but a different quantity: how high the object rises, not how far it lands. The flight time t = 2 v₀ sin(α)/g is the time from launch until the return to the launch level. The calculator reports Z as the main result, with H_max and t beside it.

History: Galileo and the parabolic path

Niccolò Tartaglia, in Nova Scientia (1537), described cannon shots and noted that an intermediate elevation, near 45°, reaches farther than a very flat or a very steep shot. Galileo gave the mathematical shape of the path in Discorsi e dimostrazioni matematiche intorno a due nuove scienze (1638), on the Fourth Day: horizontal motion is uniform, vertical motion is uniformly accelerated, and the combination is a parabola.

The form Z = v₀² sin(2α)/g is a later trigonometric writing of that same result. Sine, cosine, and the constant g in today’s notation do not appear in Galileo in this form. Mechanics textbooks (Halliday, Resnick, and Walker; Young and Freedman) derive the formula from Newton’s second law at constant g and with zero drag. The SI defines the meter and the second (BIPM, SI Brochure); the formula itself is a classroom result, not a statutory standard.

Derivation of Z = v₀² sin(2α)/g

Step 1. Split the launch speed: vₓ = v₀ cos α stays constant, and vᵧ = v₀ sin α decreases by g each second. Step 2. The vertical coordinate returns to zero when vᵧ − g t = −vᵧ, so t = 2 v₀ sin α / g. The other root, t = 0, is the launch instant. Step 3. Range is the horizontal distance in that time: Z = vₓ · t = v₀ cos α · 2 v₀ sin α / g = v₀² · 2 sin α cos α / g. The identity 2 sin α cos α = sin(2α) gives Z = v₀² sin(2α)/g.

Step 4. The sine reaches 1 at 2α = 90°, so α = 45° and Z_max = v₀²/g. The angle 90° − α has the same sin(2α), because sin(180° − 2α) = sin(2α). Step 5. For v₀ = 20 m/s, α = 45°, and g = 9.81 m/s²: sin(90°) = 1, Z = 400 / 9.81 = 40.77 m, H_max = 200 / 19.62 = 10.19 m, t = 2.88 s, and vₓ = vᵧ = 14.14 m/s. At 30°, sin(60°) ≈ 0.866, so Z = 35.31 m.

Assumptions: level ground and no drag

The formula is exact only when landing is at the launch height, g is constant, and the drag force is zero. A launch from a balcony, a tower, or a table edge has a different final height and needs a quadratic equation in h₀, which the full-projectile calculator solves. A horizontal throw is the case α = 0° with h₀ greater than zero: this formula would then give Z = 0, which is false at an edge.

Air resistance grows with speed and with the object’s area. A ball, a shuttlecock, and a bullet land closer than they would in a vacuum, and the angle of greatest range usually drops below 45°. The vacuum formula at 800 m/s and 45° gives 640,000 / 9.81 ≈ 65,240 m. A real bullet in the atmosphere travels much less far. The calculator does not adjust the result for drag, wind, or spin.

Complementary angles and the maximum at 45°

Two angles that add to 90° give the same range on level ground and a different shape of path. At 20 m/s, 30° gives a peak of 5.10 m and a time of 2.04 s, while 60° gives a peak of 15.29 m and a time of 3.53 s. The range in both cases is 35.31 m. The panel shows the complementary angle so that pair is visible without a second calculation.

Speed enters as a square. At 45°, 20 m/s gives 40.77 m and 40 m/s gives 163.10 m. g sits in the denominator: the same 20 m/s at 45° with g = 1.62 m/s², the surface gravity of the Moon, gives 246.91 m. That is still a vacuum model, not a description of flight in a thin atmosphere.

Orders of magnitude: from a stone to a vacuum bullet

A stone thrown at 10 m/s and 45° lands after 10.19 m. A low throw at 12 m/s and 15° gives only 7.34 m, because sin(30°) = 0.5. A speed of 16.6 m/s at 45° gives 28.09 m, about the length of a basketball court (28 m), if drag is ignored. A kick at 28 m/s and 40° in a vacuum gives 78.70 m; in air the range of the ball is shorter. A steep throw at 15 m/s and 80° has a range of 7.84 m and a peak of 11.12 m.

At high speeds the vacuum formula quickly stops describing air. 40 m/s at 45° is 163.10 m with no drag. 800 m/s at 45° is about 65,240 m in a vacuum, while the range of a rifle bullet in the atmosphere is many times smaller. The same formula remains valid as the no-drag limit; it is not a ballistic table.

Sources

The definition, derivation, and history come from mechanics textbooks and historical sources: G. Galilei, Discorsi e dimostrazioni matematiche intorno a due nuove scienze, 1638, Fourth Day (parabolic motion); N. Tartaglia, Nova Scientia, 1537 (gun elevation); D. Halliday, R. Resnick, J. Walker, Fundamentals of Physics, the chapter on motion in two dimensions; H. D. Young and R. A. Freedman, University Physics, “Motion in Two or Three Dimensions” (projectile motion); BIPM, The International System of Units (SI), 9th edition (meter, second). This page is for educational purposes.