Projectile Motion Calculator
Calculate range, max height, flight time, and final velocity for any projectile.
🎯 What is Projectile Motion?
Projectile motion is the motion of an object launched into the air that moves under the influence of gravity alone (ignoring air resistance). Once launched, only gravity acts on the object - there is no horizontal force. This means horizontal velocity stays constant throughout the flight while vertical velocity changes at a constant rate (acceleration = g = 9.81 m/s² downward).
The path traced by a projectile is a parabola. Projectile motion problems appear throughout physics, engineering, and sports - from calculating the range of a cannonball to analysing the trajectory of a basketball shot, a ski jumper's flight, or the arc of a thrown cricket ball.
The key insight is that horizontal and vertical motion are completely independent of each other. You can analyse each direction separately using kinematics, then combine them to find the full trajectory.
📐 Projectile Motion Formulas
vₓ = v₀ × cos(θ) v_y₀ = v₀ × sin(θ)
Time of Flight: T = 2 × v_y₀ / g
Max Height: H = v_y₀² / (2g)
Range: R = vₓ × T = v₀² × sin(2θ) / g
v₀ = initial velocity (m/s)
θ = launch angle above horizontal (degrees)
g = gravitational acceleration (9.81 m/s² on Earth)
vₓ = constant horizontal velocity component
v_y₀ = initial vertical velocity component
📖 How to Use This Calculator
Steps
1
Enter the initial velocity in m/s - this is the speed at the moment of launch.
2
Set the launch angle - 0° is horizontal, 90° is straight up. For maximum range, use 45°.
3
Adjust g if needed - leave at 9.81 for Earth, change for Moon (1.62) or Mars (3.72).
4
Click Calculate to get range, max height, flight time, and velocity components.
💡 Example Calculations
Example 1 - Optimal Range (v₀ = 30 m/s, θ = 45°)
1
vₓ = 30 × cos(45°) = 30 × 0.7071 = 21.21 m/s
2
v_y₀ = 30 × sin(45°) = 21.21 m/s | T = 2 × 21.21 / 9.81 = 4.33 s
3
H = 21.21² / (2 × 9.81) = 449.66 / 19.62 = 22.92 m
Range = 21.21 × 4.33 = 91.84 m | Max Height = 22.92 m
Try this example →Example 2 - Low angle launch (v₀ = 50 m/s, θ = 20°)
1
vₓ = 50 × cos(20°) = 46.98 m/s | v_y₀ = 50 × sin(20°) = 17.10 m/s
2
T = 2 × 17.10 / 9.81 = 3.49 s | H = 17.10² / (2 × 9.81) = 14.92 m
Range = 46.98 × 3.49 = 163.9 m
Try this example →❓ Frequently Asked Questions
Why does a 45° angle give the maximum range?
Range = v₀² × sin(2θ) / g. The sine function reaches its maximum of 1 when its argument is 90°, i.e. when 2θ = 90° → θ = 45°. Any angle above or below 45° gives a smaller value of sin(2θ) and therefore a shorter range. Complementary angles like 30° and 60° give equal ranges because sin(60°) = sin(120°) = sin(180°−60°).
What is the horizontal range formula for projectile motion?
Range R = (v0 squared x sin(2 theta)) / g, where v0 is initial speed, theta is launch angle, and g = 9.8 m/s squared. Maximum range occurs at theta = 45 degrees. Doubling the initial speed quadruples the range, since v0 is squared. This calculator uses this formula with full time-of-flight and apex height output.
How do you calculate time of flight for a projectile?
Time of flight T = (2 x v0 x sin theta) / g for a projectile launched and landing at the same height. If launched from a height h, use T = [v0 sin theta + sqrt((v0 sin theta)^2 + 2gh)] / g. Enter values in this calculator to get exact time, apex height, and range simultaneously.
What is the maximum height formula for projectile motion?
Maximum height H = (v0 squared x sin squared theta) / (2g). At 45 degrees launch with v0 = 20 m/s, H = (400 x 0.5) / 19.6 = 10.2 m. Height depends on the vertical component of velocity only. The horizontal component does not contribute to maximum height.
Why does a projectile follow a parabolic path?
Gravity provides constant downward acceleration (9.8 m/s squared) while horizontal velocity remains constant (no air resistance). Constant acceleration in one direction combined with constant velocity in the perpendicular direction produces a parabola - the same curve described by y = x squared in mathematics.
Does air resistance affect real projectiles significantly?
Yes, significantly for high-speed or small/light projectiles. A bullet, shuttlecock, or golf ball experiences substantial drag that reduces both range and height compared to the theoretical vacuum values. This calculator assumes no air resistance (ideal projectile motion), which is a good approximation for dense objects at moderate speeds but becomes inaccurate for fast-moving or aerodynamically light objects.
What if the projectile is launched from an elevated position?
This calculator assumes launch from ground level (height = 0). For elevated launches, the range formula becomes more complex because the flight time depends on how far the projectile must fall before reaching the landing height. You would need to solve a quadratic equation for time using y(t) = v_y₀·t − ½g·t², setting y = −initial_height and solving for positive t, then multiply by vₓ to get range.
How does gravity on the Moon affect projectile range?
On the Moon, g = 1.62 m/s² (about 1/6 of Earth's). Since Range ∝ 1/g, the same projectile launched at the same speed and angle would travel approximately 6× farther on the Moon. Similarly, flight time and max height would both be approximately 6× greater. Change the g value in this calculator to 1.62 to see exact Moon values.
What is projectile motion?
Projectile motion describes the curved path of an object launched with an initial velocity under constant gravitational acceleration. The horizontal and vertical components are independent: horizontal velocity is constant (no air resistance), while vertical velocity changes at 9.81 m/s^2. The result is a parabolic trajectory. Examples: a thrown ball, a bullet fired horizontally, water from a hose.
At what angle is projectile range maximized?
On level ground with no air resistance, range is maximized at a 45-degree launch angle. Range = v^2 x sin(2theta) / g, where sin(2 x 45) = sin(90) = 1 (its maximum value). For the same initial speed, 30 and 60 degrees give equal range (both sin(60) = sin(120) = 0.866). With air resistance, the optimal angle is slightly less than 45 degrees.
How does air resistance affect projectile motion?
Air resistance (drag) acts opposite to the direction of motion, reducing both horizontal range and maximum height compared to the idealized no-air-resistance case. The effect is proportional to velocity squared (for most objects in air), so high-speed projectiles are affected far more than slow ones. This calculator uses the simplified model without air resistance, which is accurate for dense, low-velocity objects but less so for feathers, paper planes, or high-speed bullets.
How do I calculate projectile motion step by step?
Step 1: split initial velocity into components: vx = v0 cos(theta), vy = v0 sin(theta). Step 2: time of flight: T = 2 x vy / g. Step 3: maximum height: H = vy^2 / (2g). Step 4: horizontal range: R = vx x T. Step 5: velocity at any time: vx(t) = vx (constant), vy(t) = vy - g x t. This calculator handles all steps automatically.