Projectile Motion Calculator
Solve basic projectile motion (no air resistance) for range, maximum height, or total time of flight.
Calculation Steps
Practical Guide
Projectile Motion Calculator
This guide explains the physics behind projectile motion, how to use the calculator correctly, and how to interpret its outputs. You’ll learn what assumptions are baked into the equations, which variables matter most, and how to sanity-check results with worked examples. The calculator here models ideal projectile motion: constant gravity, no air resistance, and a flat landing surface at \(y=0\).
Quick Start
Use these steps to get a correct, meaningful result on the first try. The calculator solves standard “textbook” projectile motion with gravity only.
- 1 Choose what you want to compute in Solve For: horizontal range \(R\), maximum height \(H_{max}\), or time of flight \(T\).
- 2 Pick Output Units (metric or imperial). This only changes display; the physics is the same.
- 3 Enter the initial speed \(v_0\). Use a realistic value for the situation (sports throws are often 10–45 m/s; small-arms ballistics are far higher).
- 4 Enter the launch angle \(\theta\) above horizontal. Angles near 0° give short, flat trajectories; angles near 90° give tall, short-range arcs.
- 5 Add an initial height \(y_0\) if the projectile starts above the landing surface. Leave at 0 for ground-level launches.
- 6 Confirm gravity \(g\). If you’re on Earth, keep the default (9.81 m/s² or 32.174 ft/s²). Change only for other planets or special tests.
- 7 Read the main output, then check the Quick Stats for the other two results and the velocity components. If a number looks off, double-check units and angle.
Tip: If you entered speed in mph or km/h, make sure that unit is selected next to \(v_0\). Most “wrong answers” are unit mismatches.
Assumption check: This calculator ignores air resistance and wind. For light objects (badminton shuttlecocks, paper, wide-bladed arrows), real range will be much shorter.
Choosing Your Method
Engineers and students use a few different approaches for projectile problems. The calculator implements the most common closed-form method, but you should know when other methods are better.
Method A — Constant-\(g\) Kinematics (this calculator)
Treat motion as independent horizontal and vertical components with constant acceleration \(-g\) vertically and zero horizontally. The core equations are:
\[ x(t)=v_0\cos\theta \, t,\quad y(t)=y_0+v_0\sin\theta \, t-\frac{1}{2}gt^2 \]
- Fast and exact for ideal conditions.
- Great for homework, early design estimates, and sanity checks.
- Easy to rearrange for range, height, or time.
- Ignores drag and wind.
- Assumes flat landing at \(y=0\) and constant gravity.
Method B — Energy + Kinematics Hybrid
For maximum height and vertical behavior, many analysts use energy relationships:
\[ H_{max}=y_0+\frac{v_{0y}^2}{2g} \]
- Quick for peak height and minimum launch speed questions.
- Good when you only care about vertical performance.
- Still idealized; doesn’t help with drag-dominated ranges.
Method C — Numerical Simulation With Drag
When drag matters, you integrate:
\[ m\ddot{\mathbf{r}} = m\mathbf{g} – \frac{1}{2}\rho C_d A |\dot{\mathbf{r}}|\dot{\mathbf{r}} \]
- Matches real trajectories for projectiles moving through air.
- Handles wind, varying gravity, and non-flat terrain.
- Needs \(C_d\), area \(A\), and air density \(\rho\), which are often uncertain.
- Slower and more complex; not ideal for quick estimates.
If your object is dense and relatively streamlined (baseball, shot-put, many bullets), Method A is often accurate enough for a first pass. If it’s light, slow, or has big cross-sectional area (foam balls, badminton birdies, thrown paper), Method C is the realistic choice.
What Moves the Number the Most
Projectile motion is sensitive to a small set of inputs. These “levers” dominate the result and are worth checking first.
Range and height scale roughly with \(v_0^2\). A 10% increase in speed can raise range by about 20% in the ideal model.
For same-height launches, ideal range is proportional to \(\sin 2\theta\). Small angle errors near 45° can produce large range differences.
A higher launch increases time aloft, which increases range even if \(v_0\) is unchanged. The effect is strongest at low angles.
Time of flight scales as \(1/g\). Lower gravity (Moon, Mars) greatly increases range and height.
The calculator assumes landing at \(y=0\). If the projectile lands on a hill or platform, real time and range change.
Drag always reduces real range and peak height. The faster and lighter the object, the larger the reduction.
Engineering shortcut: For same-height launches without drag, maximum range occurs at \(\theta \approx 45^\circ\). With drag, the optimum angle is typically lower (often 30–40°).
Worked Examples
Example 1 — Metric launch from shoulder height
- Initial speed: \(v_0 = 35\ \text{m/s}\)
- Angle: \(\theta = 40^\circ\)
- Initial height: \(y_0 = 1.5\ \text{m}\)
- Gravity: \(g = 9.81\ \text{m/s}^2\)
First resolve velocity into components: \[ v_{0x}=v_0\cos\theta,\quad v_{0y}=v_0\sin\theta \] \[ v_{0x}=35\cos40^\circ \approx 26.81\ \text{m/s},\quad v_{0y}=35\sin40^\circ \approx 22.50\ \text{m/s} \]
Time of flight comes from the vertical position equation set to zero at landing: \[ 0 = y_0 + v_{0y}T – \frac{1}{2}gT^2 \] Solving for the positive root: \[ T=\frac{v_{0y}+\sqrt{v_{0y}^2+2gy_0}}{g} \] \[ T=\frac{22.50+\sqrt{(22.50)^2+2(9.81)(1.5)}}{9.81} \approx 4.65\ \text{s} \]
Range uses horizontal motion: \[ R=v_{0x}T = 26.81\times 4.65 \approx 124.74\ \text{m} \]
Maximum height occurs when vertical velocity reaches zero: \[ H_{max}=y_0+\frac{v_{0y}^2}{2g} =1.5+\frac{(22.50)^2}{2(9.81)} \approx 27.30\ \text{m} \]
Interpretation: In reality, a baseball-like object at 35 m/s will lose some range to drag, so expect the real horizontal distance to be somewhat less than ~125 m.
Example 2 — Imperial rooftop launch
- Initial speed: \(v_0 = 180\ \text{ft/s}\)
- Angle: \(\theta = 55^\circ\)
- Initial height: \(y_0 = 50\ \text{ft}\)
- Gravity: \(g = 32.174\ \text{ft/s}^2\)
Components: \[ v_{0x}=180\cos55^\circ \approx 103.24\ \text{ft/s},\quad v_{0y}=180\sin55^\circ \approx 147.45\ \text{ft/s} \]
Time of flight: \[ T=\frac{v_{0y}+\sqrt{v_{0y}^2+2gy_0}}{g} =\frac{147.45+\sqrt{(147.45)^2+2(32.174)(50)}}{32.174} \approx 9.49\ \text{s} \]
Range: \[ R=v_{0x}T =103.24\times 9.49 \approx 980.10\ \text{ft} \]
Maximum height: \[ H_{max}=y_0+\frac{v_{0y}^2}{2g} =50+\frac{(147.45)^2}{2(32.174)} \approx 387.86\ \text{ft} \]
Sanity check: A near-1,000 ft range is plausible for a fast, dense projectile in an ideal model. With air drag, you’d expect a noticeable reduction unless the object is very streamlined.
Common Layouts & Variations
Real problems often vary from the simplest “launch and land at the same height on Earth” setup. The calculator still helps, but interpret results using the right configuration mindset.
| Configuration | How to Use the Calculator | What Changes Physically |
|---|---|---|
| Same launch and landing height (\(y_0=0\)) | Set \(y_0=0\). Ideal range \(\;R=\frac{v_0^2\sin2\theta}{g}\). | Most “textbook” case; 45° gives max range without drag. |
| Elevated launch (platform, hill) | Enter \(y_0>0\). Use the time root shown in the calculator. | Extra fall time increases range; peak height rises by \(y_0\). |
| Different gravity (Moon/Mars tests) | Change \(g\) to local value in m/s² or ft/s². | Lower \(g\) ⇒ longer flight, higher apex, longer range. |
| Landing above/below ground level | This calculator assumes landing at \(y=0\). Use \(y_0\) relative to landing surface. | If landing is higher than launch, real time/range are less than the model predicts. |
| With air resistance | Use the calculator as an upper-bound estimate only. | Drag reduces range/height; optimum angle shifts lower than 45°. |
| Wind or cross-flow | Not modeled here. Run separate aerodynamic or CFD estimates if needed. | Wind can add/subtract range and shift impact point laterally. |
- Keep angle measured from horizontal, not from vertical.
- Use \(y_0\) relative to where it lands (ground, target deck, etc.).
- Range is horizontal distance, not straight-line distance traveled.
- Peak height is above the same reference as \(y_0\).
Specs, Logistics & Sanity Checks
Before you rely on a projectile estimate for design, testing, or safety work, verify inputs and assumptions. This is where most engineering errors happen.
Input realism
- Is \(v_0\) in the correct unit? (m/s vs mph is a 2.24× difference.)
- Are you using the true launch angle, not the sight angle?
- Is the initial height measured from the landing surface?
Model limitations
- No drag, no lift, no spin.
- Constant \(g\), flat terrain.
- Projectile treated as a point mass.
If your projectile has lift (e.g., a golf ball with backspin), the real range could be longer than this ideal drag-free prediction in some regimes, but typically drag dominates and shortens range.
Sanity ranges
- At fixed \(v_0\), range should increase up to ~45° then decrease.
- Time of flight should grow with larger \(y_0\) and larger \(\theta\).
- Maximum height should rise strongly with \(\theta\) and \(v_0\).
For field tests, a good workflow is to compute an ideal trajectory first, then apply a correction factor from prior experiments. For example, if similar throws land at ~80% of ideal range, multiply this calculator’s range by 0.8 for planning. Keep safety margins whenever the projectile impacts people, structures, or critical hardware.