Projectile Motion Calculator
Full trajectory analysis: range, maximum height, flight time and impact angle with a plot.
What is Projectile Motion?
Projectile motion is what you get when a launched object moves under gravity alone: horizontal velocity is constant and vertical velocity changes at g. The two directions are independent, which is the insight that makes every projectile problem two simple one-dimensional problems instead of one hard two-dimensional one.
How it works
The launch velocity is split into components (vx = v₀cosθ, vy = v₀sinθ). The time to peak comes from vy ÷ g, the maximum height includes the launch height, and the time of flight solves the quadratic for when the vertical position returns to zero — which only reduces to the familiar 2vy ÷ g when launched from ground level. The plot and the position table are both sampled from the same computed trajectory, so they cannot disagree; the plot is the Geometry category's coordinate plane, reused rather than rebuilt.
- Enter what you know. Every field recalculates as you type — there is no submit button. Leave the quantity you want blank and the page solves for it.
- Read the answer and the working. The result appears above the fold; underneath it is the formula with your numbers substituted, so the arithmetic can be checked rather than trusted.
- Check the units and the size. Units are shown on every value and selectable per field, so a result in kJ, kPa or km/h is converted rather than assumed. Sanity-check the magnitude against a known landmark such as 1 g/cm³ for water.
Examples
A textbook problem with a known answer
Every tool here is checked against standard example problems — the molar mass of C6H12O6 is 180.16 g/mol, and free fall from 20 m takes about 2.02 s at g = 9.8 m/s².
A result you want to reason backwards from
Leave the quantity you already know blank instead: given the final answer, most tools recover the starting value — the mass needed for a target concentration, or the height a fall must have had.
Common mistakes
Forgetting that a formula is only as good as its subscripts
H2O and H2O2 are different compounds. The formula field shows its parse — element by element — so a mis-typed subscript is visible before it reaches the arithmetic.
Mixing units in the middle of a calculation
The most common source of a wrong answer is a value in grams next to a formula expecting kilograms, or Celsius next to a gas constant expecting kelvin. This category converts on the way in and labels what it used.
Frequently asked questions
What launch angle gives the maximum range?
45° when launched from ground level and landing at the same height. The optimum drops below 45° as launch height increases, because the extra downward flight favours horizontal speed.
Does the horizontal velocity change?
Not in this model. With no air resistance there is no horizontal force, so vx stays constant for the whole flight while vy changes at g every second.
Why does launching from a height change the range?
The projectile stays in the air longer, so the constant horizontal velocity carries it further. The time-of-flight formula includes the launch height for exactly that reason.