Drag-free trajectory
Launch speed, angle, height and gravity produce a to-scale trajectory and the flight numbers behind it.
Calculator
Every input updates the result, visual proof, report, code snippets and printable worksheet in the browser.
Range 40.7886 m, max height 10.1972 m, time 2.88419 s.
Use it in code
import math
speed=20
angle=math.radians(45)
height=0
g=9.80665
vx=speed*math.cos(angle)
vy=speed*math.sin(angle)
time=(vy+math.sqrt(vy*vy+2*g*height))/g
range_m=vx*time
max_h=height+vy*vy/(2*g)
print(f"range={range_m:.6g}")
print(f"height={max_h:.6g}")
print(f"time={time:.6g}")The snippets use the current inputs and print the same headline result.
Print room
The worksheet records the inputs, formula, result and a short check prompt.
CalculationTime Phase 4 worksheet
Range 40.7886 m, max height 10.1972 m, time 2.88419 s.
| Input A | 20 |
|---|---|
| Input B / mode | 45 |
| Input C | 0 |
| Input D | 9.80665 |
| Range | 40.7886 m |
|---|---|
| Max height | 10.1972 m |
| Time of flight | 2.88419 s |
| Impact speed | 20 m/s |
| Impact angle | 45 deg |
Confirm units, constants, rounding and the real-world assumption before using the number outside this worksheet.
Method
Search intent: Best answer for a drag-free projectile path from speed, angle, launch height and gravity.
Nearest sibling: Different from speed or distance calculators because it solves a two-axis motion path over time.
Projectile motion splits one launch velocity into horizontal and vertical components. Horizontal speed stays constant in the drag-free model. Vertical speed changes under gravity. Those two clocks combine to determine time of flight, range, maximum height, impact speed and impact angle.
The familiar 45-degree maximum-range rule only holds for launch height zero and no drag. When the launch starts above the landing plane, the best angle moves below 45 degrees because the projectile already has extra fall time. The page includes angle-family overlays so that rule becomes visible rather than memorised.
The default Earth gravity is the standard acceleration 9.80665 m/s^2. Moon and Mars presets are scenario values for comparison. For a lab or field job, set the custom gravity and remember that air resistance can dominate small, light or fast objects.
The trajectory picture is scaled from the calculated range and maximum height, not a fixed decorative curve. That scaling matters because a low fast shot, a steep toss and a high launch platform can have very different visual shapes even when the same equations are used. The family overlay keeps the famous 45-degree rule honest by showing it as a special case rather than a universal law.
The page follows the CalculationTime five-beat pattern: name the inputs, state the intent, calculate the result, show proof, then make the output portable. The live diagram is not decoration; it is another view of the same state, so changing an input must visibly change the geometry. The code snippets are the second audit trail. They use the current numbers and standard libraries where available, which lets a reader repeat the result without trusting the browser UI alone. The report button turns the current state into plain Markdown, which is useful for classroom notes, lab records, engineering comments, tutoring answers and audit trails because it carries inputs, assumptions and results together instead of leaving a screenshot with no method. The worksheet does a different job: it slows the calculation down enough for a person to check units, copy the formula, circle assumptions and compare the output with an official document or source. Those two outputs are deliberately boring and portable because boring portable proof is what survives after the browser tab is closed.
The source boundary is also part of the design. Constants that are exact by definition, such as inch-to-metre, pound-to-kilogram, byte prefixes and standard gravity, are separated from approximations and rules of thumb. Security pages distinguish checksums, encodings and encryption because those words are often misused in search results. Physics pages name their model limits before giving a crisp answer because a clean drag-free arc or ideal energy transfer can be mathematically correct while still being incomplete for a real projectile, falling object or electrical assembly. This is why every page carries both a direct answer and a method section: the answer is fast, but the method tells the user when the answer is allowed to matter. A good calculator page also has to survive the second question, not just the first. After the headline result, users ask what changed, what was assumed, how to repeat it, and whether a neighbouring page would have answered a different intent. Phase 4 pages therefore keep the controls, visual, code, worksheet and sources close together so the user can trace the same state through several forms of proof without hunting around the site. That repeated structure is intentional: once a visitor learns where the proof, copy button and worksheet live on one technical calculator, the next technical page becomes faster to audit and harder to misuse.
Common mistakes
Every page names the units, constants and model boundary. Change those first if your real problem uses a different standard, gravity, encoding or electrical configuration.
The interface rounds for readability. Use the code panel or copied report when you need repeatable decimal places for a calculation log.
One URL should answer one intent. Use the page that matches the job: a universal converter for units, Ohm's law for circuits, hash generator for checksums, and Base64 for encoding.
Projectile motion ignores drag, hashes are not encryption, Base64 is not security, and electrical safety needs rated parts and competent review.
Sources
Questions people ask
No. These Phase 4 tools compute in the browser from local formula and reference tables.
The snippets are a second proof path. They use the current inputs and print the same headline result so the arithmetic can be checked outside the page.
Definitions such as inch, pound, atmosphere, byte prefixes and standard gravity are labelled from standards sources. Approximations are described as planning examples.
It copies Markdown with the page URL, input values, assumptions and displayed results, plus CSV for table-friendly results.
No. These pages explain and check calculations, but they are not a substitute for standards compliance, engineering review, lender documents or security practice.
Need this as JSON? See the CalculationTime API developer portal: https://byebilly.github.io/calculationtime-web/
Certification notes
Phase 4 browser-only calculator page with live proof, code snippets, report copy and worksheet. Reviewed 2026-09-22.
The page is an educational browser calculator. It does not replace standards documents, engineering judgement, financial documents, security review or professional advice.
Use the canonical URL, the page title “Projectile Motion Calculator - CalculationTime”, and the review date 2026-09-22.