Is 45° really the angle for maximum range?

Only when you launch and land at the same height (h = 0). On flat ground, range R = v₀²·sin(2θ)/g peaks at sin(2θ) = 1, i.e. θ = 45°, and 30° gives the same range as 60° because they are symmetric about 45°. This calculator confirms that: set h = 0 and sweep the angle — 45° wins. But the moment you add a launch height, 45° stops being optimal (see the next question).

What is the best angle when I launch from a height?

Less than 45°, and it drops as the height grows. Throwing from a cliff or a raised platform means gravity has extra time to carry the projectile forward, so a flatter, faster-horizontal shot covers more ground. The optimal angle is θ = arctan(v₀ / √(v₀² + 2gh)). For v₀ = 20 m/s off a 10 m ledge on Earth, that is about 38°, not 45°. Try a few angles in the calculator with h set and watch the range peak shift left of 45°.

How much does ignoring air resistance change the answer?

For a dense, slow object (a shot put, a thrown rock) over short distances, very little — often within a few percent, which is why textbooks use the vacuum model. For a light or fast object (a ping-pong ball, a long-range bullet, a golf drive), air resistance matters a lot and the real range is shorter, sometimes by half. This tool gives the idealised vacuum result, which is the correct baseline and the answer your physics homework expects. Treat it as an upper bound on real-world range.

How is the impact speed calculated?

The horizontal velocity vₓ never changes (no horizontal force in the vacuum model), but the vertical velocity gains speed on the way down. At impact the vertical component is v_y − g·t (negative because it points downward), and the impact speed is the magnitude √(vₓ² + (v_y − g·t)²). For a ground-to-ground launch this equals the launch speed exactly — energy is conserved. Launch from a height and it lands faster, because the projectile also converted the potential energy of that height into speed.

Why does the trajectory look like a parabola?

Because horizontal position grows linearly with time (x = vₓ·t) while vertical position is quadratic (y = h + v_y·t − ½·g·t²). Eliminate t and y becomes a quadratic in x — the definition of a parabola. The plot here samples 80 points along that curve and draws it to scale, marking the apex (highest point) and the landing point so the geometry is obvious at a glance.

Check a physics homework answer

Your kinematics problem set asks for the range of a ball kicked at 25 m/s at 35° from the ground. You plug in v₀ = 25, θ = 35, h = 0, g = 9.81 and read the range, max height, and flight time straight off — then compare against the answer you worked out by hand. When they disagree, the trajectory plot and the v_y/vₓ breakdown make it obvious whether you mis-split the velocity or dropped a factor of two in the flight-time formula.

Tune a projectile in a 2D game

You are coding a slingshot or cannon in a 2D game and need the launch velocity that lands a shot on a target 60 m away. Set h to your muzzle height, try a couple of angles, and read the range until it matches 60 m — then copy v₀ and θ straight into your game code. The impact angle tells you how steeply the projectile arrives, useful for deciding whether it should bounce or stick.

Estimate a sports throw

A PE teacher wants to show students why a shot put released higher and flatter beats a high lob. Enter the release speed and a 2 m release height, then compare 45° against 40° and 35°. With the launch height set, the flatter angle wins on range — a concrete demonstration that the "always throw at 45°" rule only holds for ground-level launches.

Size a fountain or water jet

A landscape designer needs a water arc to clear a 3 m gap and peak around 2 m. Enter the nozzle speed and angle, read the max height and range, and adjust the angle until the arc fits — the SVG plot shows the shape so you can see the water clears the obstacle, not just that the numbers say it does.

Compare gravity across worlds

A space-science lesson asks "how far would this same jump go on Mars?" Enter the astronaut's takeoff speed and angle on Earth, then switch the gravity preset to Mars and Moon and watch the range and hang time grow. One screen makes the inverse relationship between gravity and range tangible without a single hand calculation.

Projectile Motion Calculator — flight time, range, max height & impact speed

Solve flight time, max height, range & impact speed from launch speed, angle and height — with a live trajectory plot, browser-only

Runs locally
Category Calculator
Best for Getting a realistic range before a purchase, plan, workout, or schedule decision.

Initial speed v₀ (m/s)

Launch angle θ (degrees, 0–90)

Launch height h (m)

Gravity g (m/s²)

Gravity preset

Results

Time of flight2.88 s

Maximum height10.19 m

Horizontal range40.77 m

Impact speed20 m/s

Horizontal velocity vₓ14.14 m/s

Vertical launch velocity v_y14.14 m/s

Impact angle (below horizontal)45°

Trajectory

What this tool does

A projectile motion solver that uses the full kinematic equations, including a non-zero launch height — so a ball thrown off a 10 m cliff lands later and farther than one thrown from the ground, exactly as physics says it should. Enter the initial speed v₀, launch angle θ (0–90°), launch height h, and gravity g, and it returns the time of flight, maximum height, horizontal range, impact speed, and the impact angle below horizontal. It splits v₀ into its horizontal and vertical components (vₓ = v₀·cosθ, v_y = v₀·sinθ), solves the quadratic h + v_y·t − ½·g·t² = 0 for the positive root to get flight time, then derives everything else from there. A live inline SVG draws the parabola to scale and marks the apex and landing point, so you see the shape of the shot, not just numbers. Gravity presets for the Moon (1.62), Mars (3.72), and Jupiter (24.79) let you watch the same throw stretch six times farther in lunar gravity. Every shot is encoded in a shareable URL, results copy with one click, and all the math runs in your browser — no server, no API, no waiting. The model assumes a vacuum (no air resistance), which is the standard textbook idealisation and accurate enough for homework, game ballistics, and back-of-envelope engineering estimates.

Tool details

Input: Text + Numbers; The page exposes text boxes, numeric controls, file pickers, or structured inputs depending on the tool.
Output: Live result + Copy + Preview; The result area focuses on usable output, with copy, download, or preview actions when supported.
Privacy: Browser-side processing; The main tool logic does not call an external API, so inputs normally stay in the current tab.
Save / share: Shareable URL state; Key settings are encoded in the URL so another person can reopen the same setup.
Performance budget: Initial JS <= 12 KB; No WASM budget is declared, keeping the tool quick to open on mobile.
Best fit: Calculator · Student; Category and role tags drive related tools, internal links, and quick fit checks.

How to use

1. Input

Paste or drop your content into the tool panel.
2. Process

Click the button. All processing is local in your browser.
3. Copy / Download

Copy the result or download to disk in one click.

How Projectile Motion Calculator fits into your work

Use it for fast estimates, comparisons, and planning numbers before you make the final call.

Calculation jobs

Getting a realistic range before a purchase, plan, workout, or schedule decision.
Comparing scenarios by changing one input at a time.
Turning rough assumptions into a number you can discuss.

Calculation checks

Double-check units, dates, rates, and rounding assumptions.
Treat health, finance, tax, and legal outputs as planning aids, not professional advice.
Save the inputs that produced an important result so you can reproduce it later.

Good next steps

These links move the current task into a more complete workflow.

Real-world use cases

Check a physics homework answer
Your kinematics problem set asks for the range of a ball kicked at 25 m/s at 35° from the ground. You plug in v₀ = 25, θ = 35, h = 0, g = 9.81 and read the range, max height, and flight time straight off — then compare against the answer you worked out by hand. When they disagree, the trajectory plot and the v_y/vₓ breakdown make it obvious whether you mis-split the velocity or dropped a factor of two in the flight-time formula.
Tune a projectile in a 2D game
You are coding a slingshot or cannon in a 2D game and need the launch velocity that lands a shot on a target 60 m away. Set h to your muzzle height, try a couple of angles, and read the range until it matches 60 m — then copy v₀ and θ straight into your game code. The impact angle tells you how steeply the projectile arrives, useful for deciding whether it should bounce or stick.
Estimate a sports throw
A PE teacher wants to show students why a shot put released higher and flatter beats a high lob. Enter the release speed and a 2 m release height, then compare 45° against 40° and 35°. With the launch height set, the flatter angle wins on range — a concrete demonstration that the "always throw at 45°" rule only holds for ground-level launches.
Size a fountain or water jet
A landscape designer needs a water arc to clear a 3 m gap and peak around 2 m. Enter the nozzle speed and angle, read the max height and range, and adjust the angle until the arc fits — the SVG plot shows the shape so you can see the water clears the obstacle, not just that the numbers say it does.
Compare gravity across worlds
A space-science lesson asks "how far would this same jump go on Mars?" Enter the astronaut's takeoff speed and angle on Earth, then switch the gravity preset to Mars and Moon and watch the range and hang time grow. One screen makes the inverse relationship between gravity and range tangible without a single hand calculation.

Common pitfalls

Assuming 45° always gives the maximum range. That is only true when launch and landing heights are equal. With any launch height, the best angle is less than 45° — flatter the higher you start.
Forgetting that horizontal velocity never changes. There is no horizontal force in the vacuum model, so vₓ at impact equals vₓ at launch; only the vertical component grows on the way down.
Mixing up degrees and radians. This tool takes the angle in degrees (0–90). If you compute by hand with a calculator in radian mode you will get nonsense — set the angle as degrees and convert internally.

Privacy

Every calculation — velocity decomposition, the flight-time quadratic, the range and impact-speed formulas, and the trajectory sampling for the plot — is plain JavaScript that runs in your browser tab. Nothing you enter is sent anywhere, there is no logging, and there is no external API call. The one thing to know: the shareable URL encodes your inputs in the query string (e.g. ?v0=20&angle=45&h=0&g=9.81), so a "share link" will let the destination see those numbers. For physics homework that is harmless; just avoid sharing the URL if the values are somehow sensitive.

FAQ

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Projectile Motion Calculator — flight time, range, max height & impact speed

Results

Trajectory

What this tool does

Tool details

How to use

1. Input

2. Process

3. Copy / Download

How Projectile Motion Calculator fits into your work

Calculation jobs

Calculation checks

Good next steps

Real-world use cases

Check a physics homework answer

Tune a projectile in a 2D game

Estimate a sports throw

Size a fountain or water jet

Compare gravity across worlds

Common pitfalls

Privacy

FAQ

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