m
Enter a valid height greater than zero.
Height above the landing surface
m/s
Enter a valid speed greater than zero.
Initial horizontal velocity (no vertical component)
💡 Key formulas:
t = √(2h/g) · x = v₀ × t
vᵧ = g × t · v_impact = √(v₀² + vᵧ²)
Horizontal and vertical motions are independent.
g = 9.80665 m/s² (NIST)
t = √(2h/g) · x = v₀ × t
vᵧ = g × t · v_impact = √(v₀² + vᵧ²)
Horizontal and vertical motions are independent.
g = 9.80665 m/s² (NIST)
Horizontal Range
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Sources & Methodology
Classical kinematics equations verified against NIST standard gravity and University Physics textbook formulations.
NIST CODATA — Standard Acceleration of Gravity
g₀ = 9.80665 m/s² (exact NIST definition) used in all projectile calculations on this page.
The Physics Classroom — Horizontally Launched Projectiles
Reference for horizontal projectile motion equations, independence of horizontal and vertical motion, and worked examples.
Time of flight: t = √(2h/g) — depends only on height h
Horizontal range: x = v₀ × t = v₀ × √(2h/g)
Vertical velocity at impact: vᵧ = g × t = √(2gh)
Impact speed: v = √(v₀² + vᵧ²)
Impact angle below horizontal: θ = arctan(vᵧ/v₀)
g = 9.80665 m/s² (NIST) · Air resistance not modelled
Horizontal range: x = v₀ × t = v₀ × √(2h/g)
Vertical velocity at impact: vᵧ = g × t = √(2gh)
Impact speed: v = √(v₀² + vᵧ²)
Impact angle below horizontal: θ = arctan(vᵧ/v₀)
g = 9.80665 m/s² (NIST) · Air resistance not modelled
⏱ Last reviewed: April 2026
How to Calculate Horizontal Projectile Motion in 2026
In horizontal projectile motion, an object is launched horizontally from a height with zero initial vertical velocity. Gravity acts only on the vertical component, causing the object to accelerate downward while its horizontal velocity remains constant (assuming no air resistance). The result is a parabolic trajectory.
The fundamental insight of projectile motion — first demonstrated by Galileo — is that horizontal and vertical motions are completely independent. A ball rolled off a table and one dropped straight down from the same height will both hit the floor at exactly the same time, regardless of how fast the horizontal ball is moving.
The Horizontal Projectile Equations
t = √(2h/g) x = v₀×t vᵧ = g×t v = √(v₀²+vᵧ²)
h = launch height (m) v₀ = horizontal speed (m/s) g = 9.80665 m/s²
Example — Ball rolled off a 5 m table at 10 m/s:
t = √(2×5/9.80665) = √1.019 = 1.010 s
Range x = 10 × 1.010 = 10.10 m
Vertical velocity vᵧ = 9.80665 × 1.010 = 9.90 m/s
Impact speed v = √(100 + 98.0) = √198 = 14.07 m/s
Impact angle = arctan(9.90/10) = 44.7° below horizontal
Example — Ball rolled off a 5 m table at 10 m/s:
t = √(2×5/9.80665) = √1.019 = 1.010 s
Range x = 10 × 1.010 = 10.10 m
Vertical velocity vᵧ = 9.80665 × 1.010 = 9.90 m/s
Impact speed v = √(100 + 98.0) = √198 = 14.07 m/s
Impact angle = arctan(9.90/10) = 44.7° below horizontal
Projectile Range Reference Table
| Height | H. Speed | Time of Flight | Range | Impact Speed |
|---|---|---|---|---|
| 1 m | 5 m/s | 0.451 s | 2.26 m | 6.80 m/s |
| 1 m | 10 m/s | 0.451 s | 4.52 m | 10.97 m/s |
| 5 m | 10 m/s | 1.010 s | 10.10 m | 14.07 m/s |
| 10 m | 15 m/s | 1.428 s | 21.4 m | 20.33 m/s |
| 100 m | 20 m/s | 4.515 s | 90.3 m | 47.33 m/s |
💡 Independence of horizontal and vertical motion: Horizontal speed has no effect on time of flight — only height determines how long the projectile falls. Doubling the horizontal speed doubles the range without changing the time. Doubling the height multiplies range by √2 (1.414) and increases impact vertical speed by the same factor. This independence was first proven by Galileo and confirmed by countless experiments.
Frequently Asked Questions
Horizontal projectile motion is the motion of an object launched horizontally from a height with no initial vertical velocity. The horizontal velocity stays constant (no air resistance), while gravity accelerates the object downward. The path traced is a parabola. Classic examples: a ball rolled off a table, water from a horizontal pipe, and a stone thrown horizontally from a cliff.
Time of flight: t = sqrt(2h divided by g). Range: x = v0 times t = v0 times sqrt(2h/g). Vertical velocity at impact: vy = g times t = sqrt(2gh). Impact speed: v = sqrt(v0 squared plus vy squared). Impact angle below horizontal: theta = arctan(vy divided by v0). Where h is initial height, v0 is horizontal speed, and g = 9.80665 m/s squared (NIST).
Range x = v0 times sqrt(2h/g), so range is proportional to sqrt(h). Doubling height multiplies range by sqrt(2) = 1.414, not 2. To double range by changing height alone, you must quadruple the height. This is because time of flight scales as sqrt(h), and range = speed times time.
No. Time of flight t = sqrt(2h/g) depends only on height and gravity. A faster ball takes exactly the same time to hit the ground as a slower one launched from the same height. This is the key insight of projectile motion — horizontal and vertical motions are completely independent. Horizontal speed only affects how far the projectile travels during that fixed time.
Impact speed v = sqrt(v0 squared plus vy squared), where vy = sqrt(2gh) is the vertical speed gained during the fall and v0 is the unchanged horizontal speed. The impact angle below horizontal = arctan(vy divided by v0). A projectile launched from greater height hits the ground faster and at a steeper angle, even if horizontal speed is unchanged.
Horizontal position: x = v0 times t (linear in time). Vertical position: y = h minus 0.5 times g times t squared (quadratic in time). Eliminating t: y = h minus g times x squared divided by (2 times v0 squared) — a parabola. The parabolic shape results directly from constant horizontal velocity combined with constant vertical gravitational acceleration.
Horizontal launch: initial vertical velocity = 0, launched from height h. Time of flight = sqrt(2h/g). Angled launch from ground: initial vertical velocity = v0 times sin(theta). Time of flight = 2 times v0 times sin(theta) divided by g. Range = v0 squared times sin(2 theta) divided by g. Maximum range at 45 degrees. This calculator covers the horizontal launch case only.
Yes, in the real world. Air resistance reduces both horizontal and vertical velocity. The actual range is shorter, impact speed is lower, and the path is asymmetric (steeper descent than ascent). This calculator assumes vacuum conditions — the ideal case used in introductory physics. For real ballistics, sports, and engineering, aerodynamic drag requires numerical integration.
g = 9.80665 m/s squared — the NIST standard acceleration of gravity. Actual local g varies by latitude (equator 9.780, poles 9.832 m/s squared) and altitude (decreases by about 0.003 m/s squared per 1000 m). For most physics problems and engineering at sea level, 9.80665 m/s squared gives accuracy well within 0.5 percent of local conditions.
Basketball passes and free throws follow a parabolic arc. Objects rolling off edges, cargo drops from aircraft, water jets from horizontal pipes, and ball sports all involve projectile motion. Engineers use these equations to predict impact zones, design drainage systems, plan cargo delivery, and calculate bullet drop at long ranges. Galileo's original work in projectile motion laid the foundation for Newton's laws and classical mechanics.
From impact speed v and impact angle theta, recover horizontal speed: v0 = v times cos(theta). Vertical impact speed: vy = v times sin(theta) = sqrt(2gh). Time of flight: t = vy divided by g = sqrt(2h/g). Range: x = v0 times t. Alternatively, if you know impact speed and height, first find vy = sqrt(2gh), then v0 = sqrt(v squared minus vy squared), and range = v0 times sqrt(2h/g).
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