What if the projectile lands at a different height than it was launched?

The simple R = v²·sin(2θ)/g formula assumes launch and landing heights are equal. If the landing height differs (for example, launching from a hill or shooting to a higher or lower target), you need to use the full kinematic equations for vertical motion, solve for time of flight, and then compute horizontal distance.

How accurate is this model for real sports or ballistics?

It’s a good conceptual baseline, but real projectiles experience air resistance, spin, lift, and sometimes changing elevation. Those factors typically reduce range compared to this ideal model—often by a large margin at high speed or with low‑mass objects.

Can I use this for different planets or moons?

Yes, if you adjust g in the underlying calculation or re‑run the math separately. Replace 9.81 m/s² with the gravitational acceleration for the body you care about (for example, ~1.62 m/s² for the Moon, ~3.71 m/s² for Mars).

Does this account for Coriolis effect or Earth’s rotation?

No. It’s a simple flat‑Earth, constant‑gravity model without Coriolis deflection. For long‑range artillery or ballistic problems where those effects matter, you’ll need a more sophisticated trajectory model.

science calculator

Projectile Range Calculator

Compute horizontal range of an ideal projectile given velocity and launch angle.

Results

Range (m): 91.77

How to use this calculator

Enter the projectile’s launch velocity in meters per second (m/s)—for example, the exit speed of a ball, projectile, or object tossed from ground level.
Enter the launch angle in degrees above the horizontal (0° is purely horizontal, 90° is straight up).
We compute R = v² × sin(2θ) ÷ g using g ≈ 9.81 m/s² for Earth’s surface gravity.
Review the resulting horizontal range in meters, and use it as an ideal reference before considering air resistance and height differences.
Optionally, change speed, angle, or gravity (in code) to explore how range shifts for different launch conditions or planetary bodies.

Inputs explained

Launch velocity (m/s): The initial speed of the projectile at the instant of launch, measured in meters per second. Convert from km/h or mph if needed (1 m/s ≈ 3.6 km/h ≈ 2.237 mph).
Launch angle (degrees): The angle above the horizontal at launch. On level ground with no air resistance, 45° maximizes horizontal range. Complementary angles (e.g., 30° and 60°) produce the same ideal range.

How it works

For a projectile launched from and landing at the same height with no air resistance, the horizontal range R on level ground is given by R = v² × sin(2θ) ÷ g.

Here v is the launch speed, θ is the launch angle relative to the horizontal, and g is the acceleration due to gravity (≈ 9.81 m/s² on Earth).

The sin(2θ) factor arises from combining horizontal and vertical motion: horizontal speed is v cosθ, while time aloft depends on the vertical component v sinθ and gravity.

This formula reveals several classic results: range scales with v² (doubling speed quadruples range) and is maximized when sin(2θ) is 1, which occurs at θ = 45°.

Internally, we convert your angle from degrees to radians for the sine function, apply the formula with g ≈ 9.81 m/s², and report the computed range in meters.

Formula

R = v² × sin(2θ) / g, with g ≈ 9.81 m/s² on Earth

When to use it

Solving physics homework or exam problems about projectile motion on level ground with no air resistance.
Getting a baseline range estimate for sports trajectories (kicks, throws, launches) before layering on drag, spin, or height differences.
Exploring how changes in launch angle or speed affect range, and visualizing why 45° is special in the ideal case.
Studying how range changes on planets or moons with different gravitational acceleration by adjusting g in code or parallel calculations.

Tips & cautions

Keep angles between 0° and 90° for meaningful trajectories; angles outside this range correspond to unrealistic or downward launches in this model.
Remember that air resistance, spin, lift, and elevation differences can dramatically change real‑world range—this is a best‑case, drag‑free scenario.
For launches from or to different heights (like a cliff or elevated platform), you’ll need the full projectile motion equations, not just the simple range formula.
Use the range together with v and θ to back out time of flight and maximum height if you’re comfortable with the full set of kinematic equations.
Ignores air drag, lift forces, wind, spin, and any aerodynamic effects, so real trajectories (especially fast or light projectiles) will have shorter ranges.
Assumes the projectile lands at the same vertical height it was launched from; launching from or to different elevations requires more detailed modeling.
Uses a single constant value of g (≈ 9.81 m/s²); variations with altitude or different planetary bodies are not modeled unless you adjust g in code.
Outputs range only; time of flight and maximum height are not reported directly, though they can be derived from v and θ using standard formulas.

Worked examples

Example 1: 30 m/s at 45° on Earth

Launch velocity v = 30 m/s, angle θ = 45°.
sin(2θ) = sin(90°) = 1.
R = v²/g = 30² ÷ 9.81 ≈ 900 ÷ 9.81 ≈ 91.7 m.
Interpretation: under ideal conditions, a 30 m/s launch at 45° travels just under 92 meters.

Example 2: 20 m/s at 30°

v = 20 m/s, θ = 30° → 2θ = 60°.
sin(60°) ≈ 0.866.
R ≈ (20² × 0.866) ÷ 9.81 = (400 × 0.866) ÷ 9.81 ≈ 346.4 ÷ 9.81 ≈ 35.3 m.
Note: launching at 60° would give the same ideal range because sin(120°) = sin(60°).

Example 3: Same launch on the Moon

Take v = 30 m/s, θ = 45°, but set g ≈ 1.62 m/s² for the Moon (in code or a side calculation).
R_moon = v²/g_moon = 900 ÷ 1.62 ≈ 555.6 m.
Interpretation: with lower gravity, the same launch would travel roughly six times as far on the Moon as on Earth in this idealized model.

Deep dive

Compute ideal projectile range from launch speed and angle using the classic R = v²·sin(2θ)/g formula for quick physics checks and classroom demonstrations.

Enter velocity and angle to see horizontal range on level ground, then use the result as a baseline before adding air resistance or altitude differences in more advanced models.

Great for students, teachers, and hobbyists who want a fast, intuitive way to explore how launch conditions affect projectile range in simple kinematics.

FAQs

What if the projectile lands at a different height than it was launched?: The simple R = v²·sin(2θ)/g formula assumes launch and landing heights are equal. If the landing height differs (for example, launching from a hill or shooting to a higher or lower target), you need to use the full kinematic equations for vertical motion, solve for time of flight, and then compute horizontal distance.
How accurate is this model for real sports or ballistics?: It’s a good conceptual baseline, but real projectiles experience air resistance, spin, lift, and sometimes changing elevation. Those factors typically reduce range compared to this ideal model—often by a large margin at high speed or with low‑mass objects.
Can I use this for different planets or moons?: Yes, if you adjust g in the underlying calculation or re‑run the math separately. Replace 9.81 m/s² with the gravitational acceleration for the body you care about (for example, ~1.62 m/s² for the Moon, ~3.71 m/s² for Mars).
Does this account for Coriolis effect or Earth’s rotation?: No. It’s a simple flat‑Earth, constant‑gravity model without Coriolis deflection. For long‑range artillery or ballistic problems where those effects matter, you’ll need a more sophisticated trajectory model.

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This projectile range calculator uses the idealized constant-gravity, no-drag formula R = v²·sin(2θ)/g for educational and preliminary analysis. It does not account for air resistance, spin, lift, height differences, Coriolis effects, or complex ballistics, and should not be used for safety-critical or real-world firing solutions. Always use appropriate engineering tools and safety margins for practical applications.