At what angle does a projectile achieve maximum horizontal range?

On a flat horizontal plane without air resistance, a launch angle of 45 degrees ($45^\circ$) produces the absolute maximum horizontal range.

What is the formula for maximum height of a projectile?

Maximum Height `H = (v₀² × sin²θ) / (2g)`, where v₀ is initial velocity, θ is launch angle, and g is gravitational acceleration (9.81 m/s²).

What is the formula for total time of flight?

Time of Flight `T = (2 × v₀ × sinθ) / g` when launched from and landing on the same elevation.

How does initial height affect projectile range and optimal angle?

When launching from an elevated platform ($y_0 > 0$), the optimal launch angle drops below 45 degrees, and total flight time increases due to the extra downward vertical fall.

Does projectile mass affect its trajectory in a vacuum?

No, in classical kinematics without air resistance, all objects accelerate at the identical rate ($g$) regardless of mass.

Science & Engineering Cluster

Projectile Motion Calculator (Trajectory, Flight Time & Range)

Calculate kinematic trajectory, maximum height, flight duration, and range for 2D projectiles.

TL;DR: Projectile Motion Calculator solves classical two-dimensional parabolic kinematics, computing maximum apex height ($H$), time of flight ($T$), horizontal distance range ($R$), and landing velocity.

Initial velocity (m/s)

Launch angle (degrees)

Initial height (m)

How Do You Calculate Maximum Height, Flight Time, and Range of a Projectile?

In projectile motion without air resistance: Total Flight Time T = (2 v₀ sin θ) / g; Maximum Apex Height H = (v₀² sin² θ) / (2g); Horizontal Range R = (v₀² sin 2θ) / g, where v₀ is initial launch velocity, θ is launch angle relative to horizontal, and g ≈ 9.80665 m/s² is acceleration due to gravity.

How to Use the Projectile Motion Calculator

Our Projectile Motion Calculator performs high-precision mathematical operations directly in your browser with zero latency and complete client-side privacy.

Enter initial launch velocity ($v_0$) in m/s, ft/s, km/h, or mph.
Enter launch angle ($ heta$) in degrees (0° to 90°).
Optionally enter initial launch height ($y_0$) above the ground plane.
Select gravitational acceleration ($g = 9.80665\text{ m/s}^2$ Earth default, or custom planet).
View real-time trajectory curves, flight time, max elevation, and total range.

Mathematical Formula & Equations

Solves ballistic trajectory range $R$, maximum apex height $H$, and total flight time $t$ with gravitational acceleration $g = 9.81\text{ m/s}^2$.

\[ R = \frac{v_0^2 \sin(2\theta)}{g}, \quad H_{\max} = \frac{v_0^2 \sin^2(\theta)}{2g}, \quad t = \frac{2 v_0 \sin(\theta)}{g} \]

Calculation Example

A projectile launched at $v_0 = 30\text{ m/s}$ at $\theta = 45^\circ$: $$\text{Range } R = \frac{30^2 \sin(90^\circ)}{9.81} = \frac{900}{9.81} = 91.74\text{ meters}$$

100% Client-Side Privacy & Data Security

All calculations, amortization schedules, variables, and sensitive numerical datasets execute 100% locally in your web browser memory. Your financial, medical, and personal values are never transmitted, logged, or uploaded to any external server.

Frequently Asked Questions

: On a flat horizontal plane without air resistance, a launch angle of 45 degrees ($45^\circ$) produces the absolute maximum horizontal range.
: Maximum Height `H = (v₀² × sin²θ) / (2g)`, where v₀ is initial velocity, θ is launch angle, and g is gravitational acceleration (9.81 m/s²).
: Time of Flight `T = (2 × v₀ × sinθ) / g` when launched from and landing on the same elevation.
: When launching from an elevated platform ($y_0 > 0$), the optimal launch angle drops below 45 degrees, and total flight time increases due to the extra downward vertical fall.
: No, in classical kinematics without air resistance, all objects accelerate at the identical rate ($g$) regardless of mass.

Cite This Calculator

Need to reference this online calculator in a financial proposal, academic paper, blog post, or classroom curriculum? Copy a citation below.

NexLove.org. (2026). Projectile Motion Calculator [Online Calculation Tool]. https://nexlove.org/tools/projectile-motion-calculator

“Projectile Motion Calculator.” NexLove.org, 2026, nexlove.org/tools/projectile-motion-calculator.

NexLove.org (2026) Projectile Motion Calculator. Available at: https://nexlove.org/tools/projectile-motion-calculator (Accessed: 2026).

@misc{nexlove_projectile_motion_calculator,
  title = {Projectile Motion Calculator},
  author = {{NexLove.org}},
  year = {2026},
  url = {https://nexlove.org/tools/projectile-motion-calculator}
}

Embed This Tool

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<iframe src="https://nexlove.org/embed/projectile-motion-calculator.html" width="100%" height="600" style="border:1px solid #e2e8f0;border-radius:12px" title="Projectile Motion Calculator — NexLove.org" loading="lazy"></iframe>

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