Mathematics · Mathematical Physics
Projectile Motion Calculator
Calculate ideal launch components, flight time, maximum height and horizontal range.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Resolve velocity: vx=20cos(45°)=14.142135623730951; vy=20sin(45°)=14.14213562373095.
- Flight time=2×14.14213562373095÷9.80665=2.8841929963302353; maximum height=10.197162129779281.
- Range=14.142135623730951×2.8841929963302353=40.78864851911713.
Understand Projectile motion
One idea, three depths
Choose how deeply to explain Projectile motion
Projectile motion: Calculate ideal launch components, flight time, maximum height and horizontal range.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Projectile motion to answer this question: calculate ideal launch components, flight time, maximum height and horizontal range? Enter Launch speed v, Launch angle θ, Gravity g; the calculator shows Flight time. For example: At 20 m/s and 45° with g=9.80665 m/s², range is about 40.79 m. The answer tells you Flight time.
Age 15Explain it to a 15-year-oldConnect it to the formula
Horizontal velocity stays constant while gravity changes vertical velocity, allowing the motion to be separated into perpendicular components. The rule is T=2v sinθ/g; H=v²sin²θ/(2g); R=v²sin2θ/g. Its input values are Launch speed v (m/s), Launch angle θ (°), Gravity g (m/s²), and the main result is Flight time. For example: At 20 m/s and 45° with g=9.80665 m/s², range is about 40.79 m.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated projectile motion relation over the valid real-number domain stated below. The implemented relation is T=2v sinθ/g; H=v²sin²θ/(2g); R=v²sin2θ/g, evaluated from Launch speed v (m/s), Launch angle θ (°), Gravity g (m/s²) to produce Flight time. Horizontal velocity stays constant while gravity changes vertical velocity, allowing the motion to be separated into perpendicular components. These formulas assume launch and landing occur at the same height with no air resistance.
Inputs and valid domain
- Launch speed v must be a finite real number, at least 0 in m/s.
- Launch angle θ must be a finite real number in °.
- Gravity g must be a finite real number, at least 0 in m/s².
Important boundary: These formulas assume launch and landing occur at the same height with no air resistance.
The formula
T=2v sinθ/g; H=v²sin²θ/(2g); R=v²sin2θ/g
How the calculator works through it
It substitutes Launch speed v, Launch angle θ, Gravity g into the formula and exposes every numerical step above. The main output is Flight time, accompanied by Maximum height, Horizontal range, Horizontal velocity.
Read the result correctly
The Flight time is the direct answer to “calculate ideal launch components, flight time, maximum height and horizontal range.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
At 20 m/s and 45° with g=9.80665 m/s², range is about 40.79 m.
Where this model stops being reliable
These formulas assume launch and landing occur at the same height with no air resistance.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Projectile motion works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Projectile motion uses T=2v sinθ/g; H=v²sin²θ/(2g); R=v²sin2θ/g. You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Ratios, units and dimensional meaning
Tracking ratios and units keeps the Projectile motion result physically interpretable instead of merely numerical.
Review this foundation about 5 min
Optional enrichment
- Vectors and physical direction
Vector language extends Projectile motion when magnitude and direction must be treated separately.
Review this foundation about 6 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read Launch speed v, Launch angle θ, Gravity g.
- Evaluate the principal relationship: T=2v sinθ/g; H=v²sin²θ/(2g); R=v²sin2θ/g.
- Return Flight time and check the domain conditions described above.
Python
from math import *
def projectile_motion(a, b, c) -> float:
return ((2.0 * (a * sin(((b * pi) / 180.0)))) / c)
assert abs(projectile_motion(20, 45, 9.80665) - 2.8841929963302353) < 1e-6 * max(1.0, abs(2.8841929963302353))
C
#include <assert.h>
#include <math.h>
double projectile_motion(double a, double b, double c) {
return ((2.0 * (a * sin(((b * 3.141592653589793) / 180.0)))) / c);
}
int main(void) {
const double expected = 2.8841929963302353;
const double actual = projectile_motion(20, 45, 9.80665);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double projectile_motion(double a, double b, double c) {
return ((2.0 * (a * std::sin(((b * std::numbers::pi) / 180.0)))) / c);
}
int main() {
constexpr double expected = 2.8841929963302353;
const double actual = projectile_motion(20, 45, 9.80665);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double projectile_motion(double a, double b, double c)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
extern sin
global projectile_motion
section .text
projectile_motion:
push rbp
mov rbp, rsp
sub rsp, 96
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd [rbp-24], xmm2
mov rax, 0x4000000000000000
movq xmm0, rax
movsd [rbp-48], xmm0
mov rax, 0x400921fb54442d18
movq xmm0, rax
movsd [rbp-88], xmm0
movsd xmm0, [rbp-16]
mulsd xmm0, [rbp-88]
movsd [rbp-80], xmm0
mov rax, 0x4066800000000000
movq xmm0, rax
movsd [rbp-96], xmm0
movsd xmm0, [rbp-80]
divsd xmm0, [rbp-96]
movsd [rbp-72], xmm0
movsd xmm0, [rbp-72]
call sin wrt ..plt
movsd [rbp-64], xmm0
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-64]
movsd [rbp-56], xmm0
movsd xmm0, [rbp-48]
mulsd xmm0, [rbp-56]
movsd [rbp-40], xmm0
movsd xmm0, [rbp-40]
divsd xmm0, [rbp-24]
movsd [rbp-32], xmm0
movsd xmm0, [rbp-32]
leave
ret
MATLAB
function result = projectile_motion(a, b, c)
result = ((2.0 * (a * sin(((b * pi) / 180.0)))) / c);
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_, c_] := ((2.0 * (a * Sin[((b * Pi) / 180.0)])) / c);
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
University Physics Volume 3
Read OpenStax University Physics: Quantum MechanicsCite this book
- APA 7
- Ling, S. J., Sanny, J., & Moebs, W. (2016). University physics volume 3. OpenStax. https://openstax.org/books/university-physics-volume-3/pages/1-introduction
- MLA 9
- Ling, Samuel J., et al. University Physics Volume 3. OpenStax, 2016, https://openstax.org/books/university-physics-volume-3/pages/1-introduction.
- Chicago author-date
- Ling, Samuel J., Jeff Sanny, and William Moebs. 2016. University Physics Volume 3. Houston, TX: OpenStax. https://openstax.org/books/university-physics-volume-3/pages/1-introduction.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Projectile Motion Calculator. MW SysArc Tools. https://math.mwsysarc.com/mathematical-physics/projectile-motion
MLA 9
MW SysArc. “Projectile Motion Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/mathematical-physics/projectile-motion. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Projectile Motion Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/mathematical-physics/projectile-motion.
Harvard
MW SysArc (2026) ‘Projectile Motion Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/mathematical-physics/projectile-motion (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_projectile_motion_2026,
author = {{MW SysArc}},
title = {Projectile Motion Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/mathematical-physics/projectile-motion},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Projectile Motion Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/mathematical-physics/projectile-motion
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Projectile motion do?
Calculate ideal launch components, flight time, maximum height and horizontal range.
How does the Projectile motion work?
The calculator applies T=2v sinθ/g; H=v²sin²θ/(2g); R=v²sin2θ/g. Horizontal velocity stays constant while gravity changes vertical velocity, allowing the motion to be separated into perpendicular components.
What can I learn from the Projectile motion?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .