Trajectory Calculator

Have a look at the flight path of the object with this trajectory calculator.

Last updated: December 27, 2025

CreatorFrank Zhao

Parameters

Velocity (v)

More info

m/s

Angle of launch (α)

More info

deg

Initial height (h₀)

More info

Horizontal velocity (vₓ)

More info

ft/s

Vertical velocity (vᵧ)

More info

ft/s

1Trajectory Formula

y = h + x\tan(\alpha) - \frac{gx^2}{2v^2\cos^2(\alpha)}

2Horizontal Velocity

v_x = v \cos(\alpha)

3Vertical Velocity

v_y = v \sin(\alpha)

4Speed From Components

v = \sqrt{v_x^2 + v_y^2}

yHeight

xDistance

vVelocity

αAngle

hInitial Height

gGravity

Quick Navigation

Projectile motion looks complicated until you split it into two simple stories: one moving sideways, one moving up and down. This guide explains what the Trajectory Calculator does, how to use it quickly, and how to sanity-check the results.

→Introduction / overview
→How to use (quick start)
→Interactive Animation & Visuals
→Step-by-step examples
→Real-world use cases
→Common scenarios
→Tips & best practices
→Calculation method / formulas
→Related concepts
→FAQs, limitations, sources

Introduction / overview

The Trajectory Calculator models the flight path of an object launched with an initial speed and angle. It helps you estimate key outcomes like how far it travels (range), how long it stays in the air (time of flight), and how high it rises (maximum height).

✅ If you’re studying physics: this is the “classic” projectile motion model with constant gravity. If you’re doing something practical: it’s a fast way to get a ballpark landing distance.

👥

Who is this for?

Students checking homework answers or building intuition for components.
Coaches, hobbyists, and makers doing “what angle should I try?” estimates.
Engineers/prototypers who need a quick, order-of-magnitude check.

For reliability, the calculator uses the standard kinematics equations with gravitational acceleration close to $g \approx 9.80665\ \mathrm{m/s^2}$ . If you only care about a single output, you may also like our Projectile Range Calculator or Time of Flight (Projectile Motion).

How to use / quick start

1Enter the launch speed $v$ and launch angle $\alpha$ . Pick units that match your context (m/s vs mph, degrees vs radians).
2If the object starts above the ground (hand release, platform, balcony), set an initial height $h_0$ .
3Optional: open “Velocity components” to view or enter $v_x$ and $v_y$ . (If you already know components, entering them is often the cleanest workflow.)
4Read the outputs and check the plot: the curve should be a smooth parabola, starting at $y=h_0$ and ending at $y=0$ .

💡

How to interpret results (quick intuition)

Bigger $v$ usually increases both time in the air and distance.
Increasing $\alpha$ increases height, but after a point can reduce range.
A positive $h_0$ gives extra hang time and usually extra range.

Interactive Animation & Visuals

Visualize the trajectory in real-time with our dynamic physics engine. Explore data points and trajectory phases.

Smart Snap & Hover

Move your mouse over the trajectory to explore precise data points.

Key Points

Start Point: Launch moment.
Apex: Peak of the curve.
Landing Point: Moment of impact.

Visual Indicators

Apex: Purple dot indicates the highest point.
Landing: Red dot indicates the landing point.

Legend Guide

Height (y)

Distance (x)

Trajectory

Apex

Landing

Speed

Angle

Start Point

Initial Height

Step-by-step example calculations

Example 1: a simple throw from hand height

Let’s launch with $v=20\ \mathrm{m/s}$ , angle $\alpha=35^{\circ}$ , and initial height $h_0=1.5\ \mathrm{m}$ .

v_x = v\cos(\alpha)

=

20\cos(35^{\circ})

\approx

16.38\ \mathrm{m/s}

v_y = v\sin(\alpha)

=

20\sin(35^{\circ})

\approx

11.48\ \mathrm{m/s}

t_f = \frac{v_y + \sqrt{v_y^2 + 2gh_0}}{g}

\approx

2.465\ \mathrm{s}

R = v_x\,t_f

\approx

16.38\times 2.465

\approx

40.4\ \mathrm{m}

Interpretation: a bit under half a football field. If you only care about the distance, you can cross-check quickly with our Projectile Range Calculator.

Example 2: start from velocity components

Suppose you measured components: $v_x=15\ \mathrm{m/s}$ , $v_y=10\ \mathrm{m/s}$ , and $h_0=0$ .

v = \sqrt{v_x^2 + v_y^2}

=

\sqrt{15^2 + 10^2}

\approx

18.03\ \mathrm{m/s}

\alpha = \arctan\left(\frac{v_y}{v_x}\right)

\approx

33.7^{\circ}

t_f = \frac{2v_y}{g}

\approx

2.039\ \mathrm{s}

,

R = v_x\,t_f \approx 30.6\ \mathrm{m}

Interpretation: using components is especially handy if you got data from a sensor or video analysis tool. If your motion is purely horizontal (so $v_y=0$ ), see our Horizontal Projectile Motion Calculator.

Real-world examples / use cases

1) Basketball arc (rough sanity check)

Background: you’re comparing two shooting forms. Inputs: $v=9\ \mathrm{m/s}$ , $\alpha=55^{\circ}$ , $h_0=2\ \mathrm{m}$ .

Result: the calculator gives a time-of-flight and range estimate; use it to compare “higher arc vs flatter arc” consistency.

2) Water fountain nozzle

Background: you want the stream to land in a pond at a certain distance. Inputs: choose $h_0$ from nozzle height and adjust $\alpha$ .

Application: try a few angles and look at the range. For distance-only iterations, the Projectile Range Calculator can be a faster comparison tool.

3) DIY launcher / maker project

Background: you’re prototyping a launcher and want a safety boundary. Inputs: measure $v$ (even roughly) and set a conservative high $\alpha$ .

Result: use the predicted maximum height $y_{\max}$ and range $R$ to set a buffer. For maximum height focus, see Maximum Height (Projectile Motion).

4) Classroom / exam checking

Background: you solved a projectile motion problem by hand. Inputs: type the same $v$ , $\alpha$ , and $h_0$ .

Application: compare your computed components $v_x$ , $v_y$ and your final range. If they disagree, it’s usually a unit/angle-mode issue.

Common scenarios / when to use

This calculator is especially useful when:

You can ignore air resistance and wind (slow/medium speeds, dense objects).
The landing height is roughly ground level (or you’re modeling to $y=0$ ).
You need a quick estimate to guide measurement or design iterations.

⚠️

It may be a poor fit when:

Air drag dominates (very fast projectiles, light objects, long distances).
The landing surface is higher/lower than your chosen zero height.
Strong wind, spin (Magnus effect), or lift is important.

Tips & best practices

📏
Be consistent with units
If you enter $v$ in mph but think in m/s, your results will look wildly off. Pick one system and stick with it.
🧠
Use components to avoid angle confusion
If you know the horizontal and vertical components, entering $v_x$ and $v_y$ can be more reliable than converting an angle.
🧾
Sanity-check with quick rules
With $h_0=0$ , time of flight should scale roughly like $t_f\propto v\sin(\alpha)$ . If it doesn’t, re-check degrees vs radians.
🧩
Combine calculators when you need a fuller picture
If you’re breaking a problem into steps, use Resultant Velocity to build $v$ from components, then use this trajectory tool for the path.

Calculation method / formula explanation

The model treats motion as two independent directions. Horizontally there is no acceleration, and vertically the acceleration is constant gravity.

Key variables

$v$ : launch speed, $\alpha$ : launch angle, $h_0$ : initial height
$v_x = v\cos(\alpha)$ , $v_y = v\sin(\alpha)$ , $g$ : gravitational acceleration

Position over time

x(t) = v\cos(\alpha)\,t

y(t) = h_0 + v\sin(\alpha)\,t - \frac{1}{2}gt^2

Trajectory shape (eliminate time)

y(x) = h_0 + x\tan(\alpha)

-

\frac{g\,x^2}{2v^2\cos^2(\alpha)}

This is a parabola. The curve in the chart should match this shape.

Time of flight (when it lands)

Set $y(t_f)=0$ and solve the quadratic.

0 = h_0 + v\sin(\alpha)\,t_f - \frac{1}{2}g t_f^2

t_f = \frac{v\sin(\alpha) + \sqrt{\left(v\sin(\alpha)\right)^2 + 2gh_0}}{g}

Range is then $R = v\cos(\alpha)\,t_f$ .

Related concepts / background info

Why “components” make everything easier

Splitting the launch speed into horizontal and vertical parts is the whole trick: $v_x$ controls how fast you move sideways, while $v_y$ controls how long you stay in the air.

v_x = v\cos(\alpha)

\quad

v_y = v\sin(\alpha)

What happens at the top of the arc?

At the peak, the vertical velocity becomes zero: $v_y(t_{\text{top}})=0$ . That’s why maximum height problems often reduce to one clean equation.

If you’re focusing on just the peak, try Maximum Height (Projectile Motion).

Frequently asked questions (FAQs)

Why is the path a parabola?

Because $x(t)$ is linear in $t$ while $y(t)$ is quadratic in $t$ . Eliminating $t$ gives a quadratic relation $y(x)$ .

What angle gives the maximum range?

On perfectly level ground with $h_0=0$ and no air resistance, the classic result is near $45^{\circ}$ . With a nonzero $h_0$ , the best angle is typically a bit lower.

Does this include air resistance?

No. The model assumes only constant gravity. Air drag can drastically reduce range, especially for light objects and high speeds.

Why do my results look wrong when I switch to radians?

A common mistake is entering degrees while the unit is set to radians. Remember: $35^{\circ} \approx 0.611\ \mathrm{rad}$ .

What if I already know the time in the air?

If you have $t_f$ , the horizontal distance is often just $R = v_x\,t_f$ . For problems that isolate time, our Time of Flight (Projectile Motion) can be a good companion.

Limitations / disclaimers

Assumes constant $g$ and no air resistance, wind, lift, or spin.
Assumes the ground is at $y=0$ . If your landing height differs, results won’t match reality.
This calculator is educational and informational — not a substitute for professional engineering or safety advice.

External references / sources

Projectile motion (overview)

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