Projectile Motion Calculator
Projectile motion describes the trajectory of an object launched into the air under the influence of gravity, with no air resistance. This is the standard kinematic model used in introductory physics. The motion separates into two independent components: a constant horizontal velocity vx = v0 * cos(theta) and a vertically accelerating component vy(t) = v0 * sin(theta) - g * t. This calculator accepts four inputs: initial velocity v0 (m/s), launch angle theta (degrees), initial height h0 above the landing surface (m, default 0), and gravitational acceleration g (m/s², default 9.81). It computes: the initial horizontal and vertical velocity components; maximum height above the ground; time of flight (using the quadratic formula when h0 is non-zero); and horizontal range. When launched from ground level (h0 = 0) and theta = 45 degrees, the range is maximised. Air resistance is not modelled; real-world trajectories will differ. All outputs update live as you type.
Range: -- m | Max height: -- m | Time of flight: -- s
Projectile motion formulas
All equations assume a flat landing surface, constant gravity, and no air resistance. The trajectory is a parabola. Angles are measured in degrees and converted to radians for trigonometric functions.
Velocity components at launch
vx = v0 × cos(θ)
vy = v0 × sin(θ)
Maximum height
H = h0 + vy² / (2 × g)
Time of flight (landed when y = 0)
When h0 = 0: T = 2 × vy / g
When h0 > 0: T = (vy + √(vy² + 2 × g × h0)) / g
Horizontal range
R = vx × T
Position at time t
x(t) = vx × t
y(t) = h0 + vy × t − 0.5 × g × t²
Worked example
v0 = 20 m/s, theta = 45 degrees, h0 = 0, g = 9.81 m/s²:
- vx = 20 × cos(45°) = 20 × 0.7071 = 14.14 m/s
- vy = 20 × sin(45°) = 20 × 0.7071 = 14.14 m/s
- H = 0 + 14.14² / (2 × 9.81) = 199.9 / 19.62 = 10.19 m
- T = 2 × 14.14 / 9.81 = 2.88 s
- R = 14.14 × 2.88 = 40.72 m
Projectile motion calculator: frequently asked questions
What is projectile motion?
Projectile motion is the curved path an object follows when launched into the air under the influence of gravity alone (air resistance ignored). The horizontal velocity component remains constant throughout the flight, while the vertical component decreases due to gravity until the object reaches maximum height, then increases downward until landing.
What launch angle gives maximum range on flat ground?
When launched from ground level (h0 = 0), the horizontal range is maximised at a launch angle of 45 degrees. This follows from the range formula R = v0² * sin(2*theta) / g, which is maximised when sin(2*theta) = 1, i.e., 2*theta = 90 degrees, so theta = 45 degrees. With a non-zero launch height the optimal angle shifts below 45 degrees.
Does this calculator account for air resistance?
No. This calculator uses the standard kinematic equations for ideal projectile motion, which assume no air resistance (drag) and a uniform gravitational field. Real-world projectiles (balls, bullets, arrows) are significantly affected by air resistance, especially at high speeds. For accurate trajectory modelling with drag, computational fluid dynamics or empirical ballistic data are required.
How is time of flight calculated when the launch height is above zero?
When the initial height h0 is greater than zero, the projectile lands when y(t) = 0. Setting h0 + vy*t - 0.5*g*t² = 0 gives a quadratic equation: 0.5*g*t² - vy*t - h0 = 0. The positive root is T = (vy + sqrt(vy² + 2*g*h0)) / g. This is the formula used in this calculator when h0 is non-zero.
What are the horizontal and vertical velocity components?
At launch, the initial velocity v0 is split into horizontal (vx) and vertical (vy) components using the launch angle theta: vx = v0 * cos(theta) and vy = v0 * sin(theta). The horizontal component vx remains constant throughout the flight (no air resistance). The vertical component changes due to gravity: vy(t) = vy - g * t.
Official sources
- NIST SP 330 (2019) "The International System of Units (SI)": NIST SP 330 PDF.
- NIST SI Unit definitions: nist.gov SI units.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology. General information only. Air resistance not modelled.