Horizontal Projectile Motion Calculator

What a Horizontal Launch Means

A horizontal launch means the object leaves with all of its speed pointing sideways and none of it pointing up or down. A ball rolled off a table, a cart leaving a ramp, water shooting out of a pipe: each one starts with a vertical velocity of zero and a horizontal velocity of whatever pushed it.

After release, gravity is the only force in the model. Horizontal speed stays constant because nothing pushes sideways, and vertical speed grows by 9.80665 m/s every second. NIST Special Publication 811 lists that figure as the standard acceleration of free fall, and it is exact by definition. This calculator uses it rather than 9.8 or 9.81, and the gap against 9.81 is 0.03 percent.

The model assumes a vacuum. Air is ignored, which is fine for a steel ball over a few meters and wrong for a beach ball over ten. A later section covers where the gap starts to show.

The Four Numbers a Drop Gives You

Four numbers come out of a level launch, and each one has its own equation. Height and launch speed are the only inputs.

Fall time: t = sqrt(2h/g). Height alone sets it. A 20 m drop takes 2.02 s whether the object leaves at 0 m/s or at 30 m/s.

Range: R = v t. Horizontal speed times fall time, because the horizontal speed never changes on the way down.

Impact speed: the vertical part is g t, which is 19.81 m/s after a 20 m fall. The total is sqrt(v squared + that squared). At 5 m/s the squares are 25 and 392.3, and 25 + 392.3 = 417.3, so the square root is 20.43 m/s.

Landing angle: the angle below horizontal whose tangent is vertical speed divided by horizontal speed. Here atan(19.81 / 5) = 75.8 degrees. A steep angle means a slow sideways launch, and a flat angle means a fast one.

Three of the four numbers depend on height. Only the range depends on launch speed, which is the part most people get backward.

Why Horizontal Speed Does Not Change the Fall Time

The two directions do not talk to each other. Gravity pulls down, and it pulls with the same strength whether the object is standing still or moving sideways at 100 m/s. A bullet fired level and a bullet dropped from the same height reach the ground at the same moment, which is the demonstration every physics class runs.

The reason sits in the vector split. Height depends on the vertical component alone, and the vertical component starts at zero in both cases. Horizontal speed decides where the object lands, not when. That is why doubling the launch speed doubles the range and leaves the clock untouched: at 20 m the fall lasts 2.02 s at every speed, so 5 m/s gives 10.10 m, 10 m/s gives 20.20 m and 20 m/s gives 40.39 m.

Two effects do break the rule at long range. The Earth curves away under a fast projectile, and the planet turns while the object flies. Both are negligible below a few kilometers and for flights shorter than a minute, so the flat earth model holds for anything you can throw.

NASA's ballistic flight notes make the same split: the vertical equation carries weight and drag, the horizontal equation carries only drag and inertia. Set drag to zero and the horizontal speed becomes a constant.

Worked Example: 20 m Cliff at 5 m/s

Take a 20 m ledge and a 5 m/s push. Fall time is the square root of 40 / 9.80665, which is 2.02 s. Range is 5 x 2.02 = 10.10 m, a bit over 33 feet.

The vertical speed at impact is 9.80665 x 2.02 = 19.81 m/s, so the total speed is 20.43 m/s and the entry angle is 75.8 degrees below horizontal. Put another way, the vertical part accounts for 97 percent of the impact speed, and the 5 m/s push barely registers at the end.

Compare the first row with the last. The ball moves 10.10 m forward while falling 20 m, and its speed climbs from 5.00 to 20.43 m/s. Most of that gain arrives in the final second, because the vertical speed keeps growing while the horizontal speed does not.

Worked Example: A Tabletop at 2 m/s

Now a 1.5 m table and a 2 m/s roll, which is the setup in most school labs. Fall time is the square root of 3 / 9.80665, or 0.553 s. Range is 2 x 0.553 = 1.106 m, just over a yard.

Vertical speed at impact is 9.80665 x 0.553 = 5.42 m/s. Total speed is the square root of 4 + 29.4, which is 5.78 m/s, and the entry angle is atan(5.42 / 2) = 69.8 degrees.

The lab version runs the equation backward. A steel ball rolls off the bench, hits paper on the floor and leaves a mark, and the distance from the bench edge to the mark is the range. Since the fall time comes from the height alone, the launch speed is v = R / t. A mark at 1.106 m on a 1.5 m bench means the ball left at 2.00 m/s.

Doubling the roll to 4 m/s moves the mark to 2.21 m and changes nothing else. The flight still lasts 0.553 s, and the impact speed only rises to 6.74 m/s.

Change the height instead. From a 0.9 m counter at 1.5 m/s the ball falls for 0.428 s and lands 0.643 m out, which is why a fast roll off a low table looks flat and a slow roll off a high one looks steep.

Reading the Trajectory Table

The calculator prints position every 0.1 s, which is fine enough to draw the path and coarse enough to read. The columns are time, forward distance, height, vertical speed and total speed.

Height falls on a square law, so the drop per tick grows. In a 20 m fall the first 0.1 s costs 0.049 m of height and the final 0.1 s costs 1.93 m. That uneven spacing is the fingerprint of constant acceleration, and it is why the path curves instead of running straight.

The forward column climbs in even steps, 0.5 m per 0.1 s at 5 m/s, because horizontal speed does not change. If that column ever speeds up or slows down in a vacuum, something is wrong with the inputs.

The angle column is the direction of travel at that instant, not the angle of the whole path. It starts at 0 degrees and reaches 75.8 degrees in the cliff example. To find the height at a spot that is not in the table, use t = x / v and then y = h - 4.9033 t squared.

That table also shows the square root in the time column. Quadrupling the height from 5 m to 20 m doubles the fall time from 1.010 s to 2.020 s, and the range doubles with it.

When Air Resistance Starts to Matter

Drag is the one assumption that bites in real life. It grows with the square of speed, so it is small at walking pace and large at highway speed, and it depends on how much air the object has to push aside.

The plain-language test is the Reynolds number, which compares the push of moving air to the stickiness of air. For a baseball 74 mm across moving at 20 m/s through air at 1.20 kg/m3 with a viscosity of 0.0000181 Pa s, the Reynolds number is 1.20 x 20 x 0.074 / 0.0000181 = 98,000. That puts the flow well into the turbulent range, where drag rises with the square of speed.

At 20 m/s the drag on a 145 g baseball works out to about 0.5 x 1.20 x 400 x 0.35 x 0.0043 = 0.36 N, against a weight of 0.145 x 9.80665 = 1.42 N. Drag is roughly a quarter of the weight at that speed, so the real ball lands a little short, a little late and a little slower than the vacuum answer.

Light, large objects are a different story. A balloon, a crumpled sheet of paper or a ping pong ball reaches terminal speed within a meter or two, and from then on it falls at a steady rate instead of speeding up. For those objects the vacuum equations overstate the impact speed by a wide margin.

A practical rule: for dense objects smaller than a fist and falls shorter than 3 seconds, the vacuum answer is within a few percent. For anything that flutters, spins or floats, treat the number as an upper bound and test it.

Mistakes That Change the Answer

Using 9.8 instead of 9.80665. The gap is 0.07 percent, smaller than the uncertainty in a height measured with a tape, but it is free to get right.

Mixing feet and meters. Convert height and speed to the same family before squaring anything, and remember that 1 ft = 0.3048 m exactly, so 20 m is 65.6 ft and 5 m/s is 16.4 ft/s.

Measuring height to the wrong surface. Height is the drop from the launch point to the landing surface, not to the floor. A ball leaving a 1.5 m table and landing on a 0.75 m crate falls 0.75 m, and the flight is 0.391 s instead of 0.553 s.

Adding the launch speed to the fall time. Speed changes the range and never the clock. It also does not add straight onto the impact speed: the two components combine as a right triangle, so 5 m/s forward plus 19.81 m/s down gives 20.43 m/s, not 24.81 m/s.

Reading the angle from the wrong axis. The calculator reports the angle below horizontal, so 75.8 degrees is a steep drop, and the same path sits 14.2 degrees from vertical.

Assuming the path is straight. It is a parabola. Over a 20 m fall the ball drops 0.049 m in the first 0.1 s and 1.93 m in the last 0.1 s, and a straight line would miss the landing point by meters.

The last table is the whole idea in one place. Every row has the same 2.02 s of flight and the same 19.81 m/s of vertical speed, and only the forward column changes with the launch.