Projectile Motion Experiment Calculator
This projectile motion experiment calculator works in two directions: enter a launch velocity and angle to predict where your projectile will land, or enter the measured range, flight time, and launch height from your experiment to determine the initial velocity. All standard outputs are covered, including horizontal range, maximum height, time of flight, and horizontal and vertical velocity components. Switch between metric and imperial units, and results update as you type.
What is projectile motion?
Projectile motion is the curved path followed by any object launched into the air under the influence of gravity alone, ignoring air resistance. Once a projectile leaves the launcher, no horizontal force acts on it, so horizontal velocity stays constant. Gravity accelerates the projectile downward at a constant rate, producing the characteristic parabolic arc. Every quantity in the trajectory, range, peak height, time of flight, impact speed, follows directly from two components: the horizontal velocity (Vx) and the initial vertical velocity (Vy0).
Key formulas
The trajectory splits into two independent motions. Horizontally, x(t) = Vx times t, where Vx = V cos(alpha) and alpha is the launch angle. Vertically, y(t) = h + Vy0 times t minus 0.5 times g times t squared, where Vy0 = V sin(alpha) and h is the initial height. Setting y(t) = 0 and solving gives the time of flight: t = (Vy0 + sqrt(Vy0 squared + 2gh)) / g. Range is then d = Vx times t. Maximum height is hmax = h + Vy0 squared / (2g). At launch height zero, the range is maximized at alpha = 45 degrees, which balances horizontal speed and air time equally.
How the experiment reverse-solve mode works
In a classroom experiment you often measure the landing distance and time of flight, then work backwards to find the launch speed. This calculator does that algebra for you. Horizontal velocity comes directly from d / t. Vertical velocity at launch is recovered from the kinematic equation for y: setting y = 0 and solving for Vy0 gives Vy0 = (0.5 times g times t squared minus h) / t. The total speed is then sqrt(Vx squared + Vy0 squared), and the angle is arctan(Vy0 / Vx). Sources of error in classroom experiments include reaction-time lag in manual stopwatch readings (typically 0.1-0.2 s), imperfect alignment of the range tape, and the fact that real projectiles experience some air resistance.
Tips for a better experiment
Use a consistent launch mechanism, such as a ball launcher with a fixed spring compression, so the initial velocity is repeatable. Conduct at least five trials per angle setting and average the results. A photogate or motion sensor removes reaction-time error from time-of-flight measurements. Mark the landing zone with carbon paper to see the scatter of your shots and estimate uncertainty. Changing launch height raises range and flight time independently of angle, which can help you isolate the effect of each variable. Finally, remember that this calculator assumes a vacuum: drag becomes significant for light, high-speed, or large-diameter projectiles.
Gravity on other worlds
| Body | g (m/s^2) | Relative to Earth |
|---|---|---|
| Moon | 1.62 | 16.5% |
| Mars | 3.72 | 37.9% |
| Venus | 8.87 | 90.5% |
| Earth | 9.81 | 100% |
| Saturn | 10.44 | 106% |
| Jupiter | 24.79 | 253% |
Change the gravitational acceleration input to simulate launches on different planets or moons.
Frequently asked questions
Why does a 45-degree launch angle give the maximum range?
When the launch height equals the landing height, range equals 2 times Vx times Vy0 divided by g. Because Vx = V cos(alpha) and Vy0 = V sin(alpha), the product Vx times Vy0 equals V squared times sin(alpha) times cos(alpha), which equals V squared times sin(2 alpha) / 2. This is maximized when sin(2 alpha) = 1, that is when 2 alpha = 90 degrees, or alpha = 45 degrees. If the launch point is above the landing point, the optimal angle drops below 45 degrees.
Does air resistance matter in a projectile experiment?
For dense, slow, spherical projectiles such as a steel ball at typical school-lab speeds, drag is small enough to ignore: the error it introduces is usually less than the measurement uncertainty from timing. However, for a ping-pong ball, a shuttlecock, or any high-speed shot over a long distance, air resistance reduces both range and peak height noticeably. This calculator assumes a vacuum, so results will be slightly optimistic when drag is significant.
How do I measure flight time accurately in a classroom?
A manual stopwatch typically has a reaction-time error of 0.1 to 0.2 seconds, which can introduce a 5-10 percent error in a short flight. For better precision, use a light gate or electronic photogate at the launch point and at the landing zone, a motion sensor, or slow-motion video analysed frame-by-frame. Averaging several trials reduces random error but not the systematic error from a biased timer.
What happens to range when I double the launch speed?
Range is proportional to the square of the launch speed (when height and angle are fixed). Doubling the speed quadruples the range. This is because both Vx and Vy0 double, so the product Vx times t doubles (t also doubles), giving four times the original distance.
Can I use this calculator for projectiles on other planets?
Yes. Change the gravitational acceleration input to the surface gravity of the planet you want. The Moon is 1.62 m/s squared, Mars is 3.72 m/s squared, and Jupiter is 24.79 m/s squared. The reference table on this page lists common values. All other formulas stay identical because horizontal motion is gravity-independent.