Projectile Motion Calculator
Calculate projectile range, time of flight, maximum height, launch speed, launch angle, velocity components, and impact speed from launch conditions.
Calculator is for informational purposes only. Terms and Conditions
Choose what to solve for
Select the unknown projectile value and a unit preset before entering the known values.
Enter the known values
The calculator uses SI internally and converts all entered units before solving.
Visual Check
The trajectory diagram updates with launch height, apex, landing point, range, and peak height.
Solution
Live result, quick checks, warnings, and full solution steps.
Quick checks
- Check—
Show solution steps See the equations, substitutions, assumptions, and result path
- Enter values to see the full solution steps and checks.
Source, Standards, and Assumptions
Calculation basis, constants, assumptions, and limitations.
Uses standard constant-acceleration projectile motion equations without air resistance.
- Assumes constant gravity and no aerodynamic drag, lift, spin, wind, or buoyancy.
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Calculator Guide
How to Use the Projectile Motion Calculator
The Projectile Motion Calculator above helps you calculate range, time of flight, maximum height, launch speed, launch angle, velocity components, and impact speed. Enter the known launch conditions, choose the units, and use the explanation below to understand the formulas, assumptions, and checks behind the result.
Projectile motion is calculated by separating the launch velocity into horizontal and vertical components. The horizontal component controls distance traveled, while the vertical component controls flight time and peak height.
Quick Answer
To calculate projectile motion, split the initial velocity into \(v_x=v_0\cos(\theta)\) and \(v_y=v_0\sin(\theta)\). Then solve the vertical motion equation for time and multiply by horizontal velocity to get range. Maximum height occurs when the vertical velocity becomes zero.
When not to rely on the simplified model
This calculator assumes ideal projectile motion with no air resistance, wind, spin, lift, or drag. Use it for education and quick estimates, but use a more detailed model for sports balls, bullets, rockets, high-speed objects, or any application where safety or design decisions depend on the result.
Inputs and Outputs Used by the Projectile Motion Calculator
The calculator uses launch speed, angle, height, landing height, and gravity to calculate the path of an object moving under constant gravity. Some solve modes reverse the problem and calculate the launch speed or angle needed to reach a target range.
| Type | Value | What It Means | Common Units |
|---|---|---|---|
| Input | Initial velocity | Launch speed at the start of motion. | m/s, ft/s, mph, km/h |
| Input | Launch angle | Angle above the horizontal direction. | degrees or radians |
| Input | Initial height | Starting elevation relative to the selected vertical datum. | m, ft, in, yd, cm |
| Input | Landing height | Elevation where the projectile reaches the target or ground. | m, ft, in, yd, cm |
| Input | Gravity | Acceleration due to gravity for Earth, Moon, Mars, or a custom value. | m/s² or ft/s² |
| Output | Range, time, height, speed, or angle | The calculated trajectory result based on the selected solve mode. | depends on solve mode |
Which solve mode should you choose?
Choose range when you know speed and angle. Choose required initial velocity when you know the target distance and launch angle. Choose launch angle when you know speed and target distance. Choose time of flight or maximum height when you need the flight duration or peak elevation instead of horizontal distance.
Projectile Motion Formula
Projectile motion is usually solved by splitting velocity into horizontal and vertical components. Horizontal motion is constant in the ideal no-drag model, while vertical motion uses constant acceleration from gravity.
Velocity Components
These components convert the launch speed and angle into horizontal and vertical motion.
Position Equations
The time \(t\) is shared by both horizontal and vertical motion.
Time of Flight With Launch and Landing Heights
This positive root gives the time when the projectile reaches the selected landing height \(y_L\). Use this instead of the flat-ground shortcut when launch and landing heights are different.
Range and Maximum Height
Range depends on both horizontal velocity and flight time. Maximum height depends on the vertical velocity component.
Flat-Ground Shortcut
Use this shortcut only when the projectile launches and lands at the same height.
Projectile motion with initial height
When launch and landing heights are different, solve the vertical position equation for the positive time value first, then calculate range using \(R=v_xt\). A higher starting height usually increases flight time and range because the projectile has farther to fall. A higher landing height usually decreases flight time and can make a target unreachable at low speed.
What the Variables Mean
Projectile motion variables describe launch conditions, position, gravity, and results. Use a consistent coordinate system so initial height and landing height are measured from the same reference level.
| Variable | Meaning | Notes |
|---|---|---|
| \(v_0\) | Initial velocity | Total launch speed before it is split into components. |
| \(\theta\) | Launch angle | Measured from the horizontal direction. |
| \(v_x\) | Horizontal velocity | Constant in the ideal no-drag model. |
| \(v_y\) | Initial vertical velocity | Changes over time because gravity acts vertically. |
| \(h_0\) | Initial height | Starting height relative to the selected datum. |
| \(y_L\) | Landing height | Height where the projectile reaches the target or ground. |
| \(g\) | Gravity | Use \(9.80665\ \text{m/s}^2\) for standard Earth gravity unless another value is intended. |
| \(R\) | Horizontal range | Horizontal distance traveled before reaching the landing height. |
How to Use the Projectile Motion Calculator
Use the calculator by choosing the unknown value, entering the known launch conditions, selecting units, and checking the trajectory outputs against the formulas and sanity checks below.
Select the solve mode
Choose whether you want range, time of flight, maximum height, required initial velocity, or launch angle. The required inputs change based on the selected solve mode.
Enter the known values
Enter launch speed, angle, target range, initial height, landing height, and gravity as needed. Use initial and landing heights from the same reference point.
Check the units
Confirm whether velocity is in m/s, ft/s, mph, or km/h. Confirm whether gravity is in m/s² or ft/s² before trusting the result.
Review the path and quick checks
Use range, peak height, time of flight, velocity components, impact speed, and impact angle to decide whether the trajectory is reasonable.
How to Interpret Projectile Motion Results
A projectile result is only useful if it matches the physical situation. A mathematically valid range or angle may still be unrealistic if air resistance, wind, spin, or incorrect height assumptions are important.
What to do with the result
Use it to check homework, compare launch angles, estimate a no-drag trajectory, or understand how velocity and angle affect range and height.
What changes the result most?
Initial velocity usually dominates the result because range and peak height depend strongly on speed. Small speed changes can cause large range changes.
Sanity check
For equal launch and landing height at \(45^\circ\), range should be close to \(v_0^2/g\). If it is not, check angle units, velocity units, and gravity.
Impact velocity check
Impact velocity combines the unchanged horizontal velocity with the final vertical velocity at the landing height. If the projectile lands at the same height it launched from and drag is ignored, the impact speed should match the initial speed.
Low, high, and impossible results
A low range often means low speed, a steep angle, or a high landing point. A very high range often means a unit mistake or unrealistic no-drag assumption. An impossible angle result means the target cannot be reached with the selected speed and height difference.
Input Checklist Before You Trust the Answer
Most wrong projectile motion answers come from unit mistakes, incorrect height references, or using the flat-ground shortcut when the launch and landing heights are different.
Velocity check
Confirm whether speed is entered as m/s, ft/s, mph, or km/h. A mph value treated as m/s will make the trajectory much too long.
Angle check
Confirm whether the angle is in degrees or radians. \(45^\circ\) and \(45\) radians are completely different values.
Height check
Use the same datum for initial height and landing height. A projectile launched from a platform is not the same as one launched from ground level.
Gravity check
Use \(9.80665\ \text{m/s}^2\) or about \(32.174\ \text{ft/s}^2\) for standard Earth gravity unless you intentionally choose another environment.
Projectile Motion Worked Example
This example uses a common flat-ground projectile problem. The projectile launches and lands at the same height, so the result can be checked with both component equations and the flat-ground shortcut.
Step 1: Resolve velocity
Step 2: Calculate time of flight
Step 3: Calculate range and peak height
Final answer
The projectile travels about 91.8 m, stays in the air about 4.33 s, and reaches a peak height of about 22.9 m. This is reasonable because, for equal launch and landing height at \(45^\circ\), the range is close to \(v_0^2/g\).
Calculator default check
This example also works as a quick check for the calculator defaults: a 30 m/s launch at \(45^\circ\) on level ground should return a range close to 91.8 m.
How to Visualize Projectile Motion
Projectile motion is easier to understand when the path is separated into horizontal and vertical behavior. The diagram uses numbered markers instead of crowded labels so the path stays readable on desktop and mobile.
Numbered markers keep the diagram readable: 1 = launch point, 2 = horizontal and vertical velocity components, 3 = apex, 4 = landing point, and 5 = horizontal range.
1. Launch point
The path begins at the initial height \(h_0\), which may be ground level, a platform, or another reference elevation.
2. Velocity components
The launch speed is split into \(v_x\) and \(v_y\). Horizontal velocity controls distance, while vertical velocity controls rise and fall.
3. Apex
The apex occurs when vertical velocity is zero. This is where the projectile reaches maximum height.
4–5. Landing and range
The projectile lands when it reaches the selected landing height \(y_L\). Horizontal range is the distance from launch to landing.
Reference Checks for Projectile Motion
Projectile motion does not have one universal “normal” range because speed, angle, height, and gravity control the result. Instead, use simple reference checks to catch impossible or suspicious answers.
Earth gravity check
Standard Earth gravity is about \(9.80665\ \text{m/s}^2\) or \(32.174\ \text{ft/s}^2\). If gravity is much smaller, the projectile will stay in the air longer.
Flat-ground 45° check
For equal launch and landing height at \(45^\circ\), range should be close to \(v_0^2/g\). This is a fast way to catch unit mistakes.
Peak height check
Maximum height should increase when vertical velocity increases. If the height decreases after increasing the launch angle, check the angle unit.
Impact speed check
If launch and landing heights are equal and drag is ignored, impact speed should equal launch speed.
Design Notes and Practical Ranges
Projectile motion formulas are useful for idealized physics, educational problems, and quick estimates. Real objects often need more than the no-drag model because shape, speed, spin, and air resistance can strongly change the path.
Best use case
Use the calculator for classroom physics, preliminary trajectory estimates, and comparing launch conditions under ideal assumptions.
Needs deeper analysis
Use a more advanced model when drag, wind, lift, spin, changing air density, or impact safety matters.
Target checks
If solving for launch angle, a target may be unreachable. That usually means the speed is too low, the range is too long, or the landing height is too high.
Why two launch angles can reach the same target
When speed and target range are fixed, a low-angle path and a high-angle path may both reach the same point. The low-angle path is flatter and faster, while the high-angle path rises higher and stays in the air longer. If the target is too far or too high, no real launch angle exists for the selected speed.
Projectile Motion Units and Conversions
Projectile motion calculations are sensitive to unit consistency. The safest workflow is to convert internally to meters, seconds, radians, m/s, and m/s², then convert the final result to the preferred display unit.
| Quantity | Conversion | Why It Matters |
|---|---|---|
| Velocity | \(1\ \text{mph}=0.44704\ \text{m/s}\) | Entering mph as m/s makes range far too large. |
| Velocity | \(1\ \text{km/h}=0.27778\ \text{m/s}\) | Useful when launch speed is entered from metric speed data. |
| Velocity | \(1\ \text{ft/s}=0.3048\ \text{m/s}\) | U.S. velocity units must match the gravity unit. |
| Distance | \(1\ \text{ft}=0.3048\ \text{m}\) | Height and range must use consistent distance conversions. |
| Gravity | \(9.80665\ \text{m/s}^2 \approx 32.174\ \text{ft/s}^2\) | Using \(9.80665\ \text{ft/s}^2\) by mistake makes the path unrealistic. |
| Angle | \(180^\circ=\pi\ \text{rad}\) | Degrees and radians are not interchangeable. |
Hidden unit trap
The most common U.S. unit error is switching distance to feet while leaving gravity as \(9.80665\) but labeling it as ft/s². Earth gravity in ft/s² is about \(32.174\), not \(9.80665\).
Projectile Range vs Time of Flight vs Maximum Height
Range, time of flight, and maximum height are related, but they answer different questions. Range describes horizontal distance, time of flight describes how long the projectile is airborne, and maximum height describes the peak vertical position.
Range
Use range when you need to know how far the projectile travels horizontally before reaching the landing height.
Time of flight
Use time of flight when you need to know how long the projectile remains in the air or when it reaches a target height.
Maximum height
Use maximum height when clearance, peak elevation, or vertical rise matters.
Common Projectile Motion Mistakes
Most projectile motion mistakes come from using the right formula under the wrong assumptions. The flat-ground shortcut is convenient, but it does not apply to every launch and landing height.
Do
- Split velocity into horizontal and vertical components first.
- Use the vertical equation to find time when heights are different.
- Check whether the answer changes logically when speed or angle changes.
- Use \(32.174\ \text{ft/s}^2\) for Earth gravity when working in feet.
Don’t
- Do not assume \(45^\circ\) always gives the longest range.
- Do not use \(R=v_0^2\sin(2\theta)/g\) when launch and landing heights differ.
- Do not mix degrees and radians.
- Do not confuse peak height above the selected datum with rise above the launch point.
- Do not ignore drag for real high-speed or lightweight objects.
Troubleshooting Unrealistic Projectile Results
If the result looks too large, too small, negative, or impossible, check the input units and the physical assumptions before changing the formula. The calculator may be doing the math correctly while the inputs describe the wrong situation.
Range is too high
Check whether mph was entered as m/s, gravity was entered too low, or air resistance should be included.
Range is too low
Check whether launch angle is too steep, speed is too low, or landing height is above the launch point.
Angle is impossible
The target may be unreachable with the selected speed. Increase launch speed, reduce target range, or lower the landing height.
Peak height looks wrong
Confirm whether the calculator reports peak height above the selected datum, not height above the launch point.
Assumptions and Limitations
The projectile motion calculator uses the standard ideal model. That model is excellent for learning the relationship between speed, angle, height, and gravity, but it is not a complete real-world trajectory simulator.
No air resistance
The calculator assumes drag is zero. Real projectiles often travel shorter distances than the ideal result.
Constant gravity
Gravity is treated as constant over the entire path. This is appropriate for ordinary classroom-scale problems.
Point-mass projectile
The model ignores shape, rotation, lift, Magnus effect, wind, and deformation.
Non-vertical range modes
Range and target calculations require horizontal motion. A perfectly vertical launch has height and time, but no horizontal range.
Key Projectile Motion Terms
These terms help connect the calculator inputs, formula, diagram, and result.
Trajectory
The path followed by the projectile as it moves horizontally and vertically.
Range
The horizontal distance from launch point to landing point.
Time of flight
The total time the projectile stays in the air before reaching the landing height.
Maximum height
The highest vertical position reached by the projectile relative to the selected datum.
Impact velocity
The speed and direction of the projectile when it reaches the landing height.
Launch angle
The angle between the initial velocity vector and the horizontal direction.
Projectile Motion Calculator FAQ
What is projectile motion?
Projectile motion is the motion of an object launched into the air where gravity is the only acceleration in the simplified model. Horizontal velocity is treated as constant, while vertical velocity changes because of gravity.
How do you calculate projectile range?
Calculate horizontal velocity with \(v_x=v_0\cos(\theta)\), solve the vertical motion equation for time of flight, then multiply \(R=v_xt\). The flat-ground shortcut \(R=v_0^2\sin(2\theta)/g\) only applies when launch and landing heights are equal.
What launch angle gives maximum range?
A \(45^\circ\) launch angle gives maximum range only for equal launch and landing height with no air resistance. If the projectile starts above the landing point, the best angle is usually less than \(45^\circ\).
How does initial height affect projectile range?
A higher initial height usually increases time of flight and range because the projectile has farther to fall before reaching the landing height. A higher landing height usually reduces time of flight and can make a target unreachable at low speed.
Does mass affect projectile motion?
In the ideal no-air-resistance model, mass does not affect projectile motion because all objects have the same gravitational acceleration. In real life, mass, shape, drag, spin, and wind can affect the path.
Does this calculator include air resistance?
No. This calculator uses the standard ideal projectile motion model with constant gravity and no air resistance, wind, lift, or spin effects.