The projectile motion calculator above solves the full two-dimensional problem rather than the textbook special case. Most projectile pages assume the object lands at exactly the height it left from, because that assumption collapses the range into a single tidy expression. Real throws, kicks, jumps and shots almost never satisfy it. A ball leaves a hand at shoulder height and lands on grass. A cannon sits on a rampart. A golf ball is struck from a tee and finishes on a green that is rarely level with it. This tool takes the launch height and the landing height as separate inputs and solves the quadratic properly, so the answer stays correct when those two numbers differ.
Arb Digital publishes free calculators that state their assumptions instead of hiding them. This one assumes a vacuum: no air resistance, no lift, no spin, no wind, and a gravitational field that is uniform over the whole flight. That is the standard idealisation used in every introductory mechanics course, and it is exact enough for short, heavy, slow projectiles. It is not exact for a badminton shuttle or a long-range shell, and the sections below explain where the departure begins and how large it gets.
What This Projectile Motion Calculator Does
You supply four things: how fast the object was moving when it was released, the angle of that release above the horizontal, the height it was released from, and the height of the surface it lands on. The calculator returns the horizontal distance covered, the total time in the air, the highest point reached, the speed at impact and the angle at which it strikes.
The headline figure is the horizontal range measured from the launch point to the landing point. The supporting grid holds the four numbers that are usually needed alongside it. Time of flight is what you need for interception and timing problems. Peak height is what you need for clearance problems — will it get over the wall, the net, the tree line. Impact speed and impact angle matter for anything involving damage, bounce, or the behaviour of a ball once it reaches the ground.
How to Use It
- Enter the launch speed and pick its unit. This is the speed of the object at release, not the speed of the thing that threw it and not an average speed over the flight. If your figure came from a radar gun, check the geometry first: a gun off the line of travel reads low.
- Set the launch angle. Measure it from the horizontal, upward positive. A downward launch from a height — a ball thrown off a balcony at a downward slant — takes a negative angle, and the calculator handles that case without complaint.
- Fill in both heights. Launch height is where the object left your hand or the muzzle, not where your feet are. Landing height is the level of the surface it will hit. Leave landing height at zero and set launch height to zero if you genuinely want the flat-ground textbook case.
- Choose the length unit and gravity. The length unit applies to both height inputs and to every distance in the results. Gravity stays at standard Earth gravity unless you change it.
- Read the note under the grid. It prints the flat-ground range alongside the real one and names the optimal angle for your particular pair of heights, which is only 45 degrees in the special case where they are equal.
The Formula: How Projectile Range Is Calculated
Split the launch velocity into components. The horizontal component is vx = v0 cos θ and the vertical component is vy = v0 sin θ. In a vacuum nothing acts horizontally, so vx never changes and the horizontal distance is simply vx multiplied by the time in the air. All the difficulty is in finding that time.
Vertically, the height at time t is y = y0 + vyt − gt2 ÷ 2. Setting that equal to the landing height y1 and writing Δy = y1 − y0 gives a quadratic whose positive root is t = [vy + √(vy2 − 2gΔy)] ÷ g. OpenStax University Physics Volume 1, section 4.3 on projectile motion, derives exactly this treatment and works an example in which a ball is caught ten metres above the point it was hit from.
Work the defaults through by hand. A launch speed of 20 m/s at 45 degrees gives vx = vy = 14.142 m/s. With a launch height of 2 m and a landing height of 0, Δy is −2 m, so the discriminant is 14.1422 + 2 × 9.80665 × 2 = 239.23 and its square root is 15.467. The flight time is (14.142 + 15.467) ÷ 9.80665 = 3.019 s, and the range is 14.142 × 3.019 = 42.70 m. The peak height is 2 + 14.1422 ÷ (2 × 9.80665) = 12.20 m above the ground. The gravity default of 9.80665 m/s² is the exact value fixed by the NIST CODATA entry for standard acceleration of gravity.
Now compare that with the flat-ground shortcut, R = v02 sin 2θ ÷ g, which returns 400 ÷ 9.80665 = 40.79 m. Two metres of launch height added nearly two metres of range, a difference of almost five per cent. That is the error a page hard-wired to the equal-height case would have handed you, and it grows quickly as the height difference grows.
Why the Equal-Height Shortcut Is the Wrong Default
The compact range formula everyone remembers is a special case, not the general law. It is exact only when the projectile finishes at the height it started. That condition is satisfied by a golf ball on a perfectly level fairway and by essentially nothing else people actually measure.
Two consequences follow. The first is the numerical one already shown: launching from above the landing surface always adds range, because the projectile gets extra time in the air on the way down. The second is more surprising. The optimal angle stops being 45 degrees. When you launch from above the target, the best angle drops below 45; the higher the launch relative to the landing, the lower the optimum. Shot putters release from around two metres above the ground and their optimal release angle sits close to 42 degrees before any aerodynamic correction. When you launch from below the target, the optimum rises above 45.
This calculator computes the true optimum for your specific pair of heights and prints it in the note, so you can see how far your entered angle sits from it. Setting the angle to that optimum and recalculating shows you exactly what the ideal launch would have achieved, which is a far more useful comparison than being told that 45 degrees is always best.
The Vacuum Assumption and When It Breaks
This tool models a vacuum trajectory. Drag is proportional to roughly the square of the speed and to the object's cross-sectional area, and inversely related to its mass, so the accuracy of the vacuum answer depends on the combination of those three things rather than on any one of them.
As a rough guide, the vacuum result is close for dense, compact, slow projectiles: a thrown stone, a shot put, a hammer throw, a short-range mortar round. It is poor for light, large, fast ones: a table tennis ball, a shuttlecock, a football struck hard, a long-range rifle bullet. A well-hit golf ball is a case where the vacuum model fails in the interesting direction — backspin generates lift that keeps it airborne longer than a vacuum calculation predicts, so the real range can exceed the ideal one.
If drag matters for your problem, the honest move is to treat the vacuum answer as an upper bound on range for most objects and then estimate the drag force separately. The drag force calculator gives the magnitude of that force at a given speed, which tells you immediately whether it is a small correction or the dominant term. When drag force at launch speed is a small fraction of the object's weight, the numbers on this page are usable. When it is comparable to the weight, they are not.
Reading Impact Speed and Impact Angle
The impact speed is not the launch speed unless the two heights are equal. In a vacuum, energy is conserved, so an object landing below its launch point arrives faster and one landing above arrives slower. The defaults show this: launched at 20 m/s from 2 m, the object lands at about 20.96 m/s, having converted 2 m of extra height into speed.
If you want the energy rather than the speed at impact, take the impact speed from the grid straight into the kinetic energy calculator with the object's mass. That pairing answers the question people usually mean when they ask how hard something hits.
Where This Sits Next to Our Other Motion Tools
Arb Digital publishes several kinematics calculators and it is worth being precise about the boundary between them, because they solve different slices of the same physics. This page is the two-dimensional case with an angled launch. The free fall calculator is the one-dimensional case with no horizontal component at all — a straight drop or a straight-up throw. The suvat calculator solves any single-axis constant-acceleration problem for whichever variable you are missing, which is the general engine underneath both.
For the components themselves, the velocity calculator handles displacement, time and average velocity, and the acceleration calculator covers the rate of change of velocity in one dimension.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Leaving launch height at zero out of habit — almost nothing is launched from ground level. Two metres of release height changes the range of a 20 m/s throw by nearly five per cent, and much more for slower throws.
- Assuming 45 degrees is always optimal — it is optimal only when launch and landing heights are equal. Launching from above the target lowers the optimum; launching from below raises it.
- Using a total speed that is not the release speed — the speed of a bat, club or foot is not the speed of the ball that leaves it. Use a measured ball speed where you have one.
- Applying vacuum results to light or very fast objects — for a shuttlecock or a bullet the drag term dominates and this model will be badly wrong, not slightly wrong.
- Mixing units between the two height fields — both heights and every distance result share one length unit. Enter a launch height in feet and a landing height in metres and the answer is meaningless.
Related Free Tools From Arb Digital
For vertical motion with no horizontal component, use the free fall calculator. For any single-axis constant-acceleration problem with a missing variable, the suvat calculator is the general solver, and the acceleration calculator and velocity calculator cover the simpler component cases. Take the impact speed here into the kinetic energy calculator for landing energy, or check whether the vacuum assumption holds with the drag force calculator. Browse everything at the free online tools hub.
Frequently Asked Questions
No. It models a vacuum trajectory with uniform gravity, which is the standard idealisation in introductory mechanics. That is accurate for dense, compact, relatively slow projectiles such as a shot put or a thrown stone, and inaccurate for light or very fast ones such as a shuttlecock or a bullet.
Because the projectile spends longer in the air before it reaches the landing level, and the horizontal velocity keeps carrying it forward for that whole time. Launching from above the target always adds range in a vacuum, and launching from below always removes some.
Only when the launch and landing heights are equal. If you launch from above the landing surface the optimal angle falls below 45 degrees, and if you launch from below it rises above. This tool prints the true optimum for the heights you entered.
It means the trajectory peaks below the level you set as the landing height, so there is no real solution. That happens when you aim at a target above you with too little vertical speed. Increase the launch speed or the angle until the peak height exceeds the landing height.
Energy is conserved in a vacuum, so height converts into speed. An object that lands below where it started arrives faster than it left, and one that lands higher arrives slower. The two speeds are equal only when the two heights are equal.
Yes. A negative angle means the object was launched downward from its starting height, which is what happens when you throw something at a downward slant off a balcony or a cliff. The quadratic still has a valid positive root and the calculator solves it normally.
Change the gravity field. The Moon is about 1.625 metres per second squared and Mars about 3.721, against standard Earth gravity of 9.80665. Lower gravity increases both flight time and range, and the vacuum assumption is genuinely accurate on the Moon because there is no atmosphere.
This tool is provided for educational and study use. It models idealised vacuum trajectories and does not account for air resistance, lift, spin, wind or terrain, so treat its output as a physics result rather than a prediction of a real flight.
