What angle gives maximum range in projectile motion?

When launched from level ground with no air resistance, a projectile achieves its maximum range at a launch angle of 45 degrees. At angles above or below 45 degrees the range is shorter. When the launch point is above the landing point, the optimal angle is slightly less than 45 degrees.

What is the formula for projectile range?

For a projectile launched from ground level at velocity v0 and angle theta, the horizontal range is R = v0 squared times sin(2 theta) divided by g, where g is 9.81 m/s2. When the initial height is not zero, you must solve the quadratic equation for time of flight first, then multiply horizontal velocity by that time.

Does air resistance affect projectile motion?

Yes. Real projectiles experience drag that reduces range and maximum height compared to the ideal values this calculator produces. The results here assume a vacuum (no air resistance) and a flat Earth, which gives good estimates for short-range, low-speed problems such as sports science and school physics experiments.

Projectile Motion Calculator

This calculator analyses the complete trajectory of a projectile launched at a given speed and angle from an initial height. Projectile motion is a form of two-dimensional kinematics where the horizontal and vertical components of motion are treated independently. Horizontally, velocity is constant because there is no air resistance in the ideal model. Vertically, the object accelerates downward at g = 9.81 m/s squared. You provide the initial speed in m/s, the launch angle in degrees measured from the horizontal, and the initial height in metres. From these three inputs the calculator returns the horizontal range in metres (distance from launch to landing), the maximum height reached above the launch point, the total time of flight in seconds, and the speed at landing. This tool is useful for physics students working through mechanics problems, sports scientists modelling ball trajectories, and engineers estimating throw or drop distances. Results assume no air resistance, a constant gravitational field of 9.81 m/s squared, and a flat surface. Real-world results will differ due to drag, spin, and terrain. For objects falling vertically from rest, use the Free Fall Calculator instead, which is a special case of this analysis where the launch angle is 90 degrees and initial horizontal speed is zero.

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Initial speed (v₀)

m/s

Launch angle (θ)

deg

Initial height (h₀)

40.77 m

horizontal range

Max height10.19 m

Time of flight2.88 s

Landing speed20.00 m/s

Horiz. velocity14.14 m/s

No air resistance assumed. Results are ideal values for planning and study purposes only.

How it works

The launch velocity is split into horizontal and vertical components: vx = v₀ cos(θ) and vy = v₀ sin(θ). Horizontal motion: x = vx × t. Vertical motion: y = h₀ + vy·t - ½g·t². Time of flight is found by solving y = 0 using the quadratic formula. Maximum height occurs when vertical velocity is zero: t_top = vy / g, giving h_max = h₀ + vy² / (2g). Landing speed combines the horizontal and vertical components at landing: v_land = sqrt(vx² + vy_land²).

Worked example

A ball is launched at v₀ = 20 m/s at θ = 45 degrees from ground level (h₀ = 0). Horizontal velocity = 20 cos(45) = 14.14 m/s. Vertical velocity = 20 sin(45) = 14.14 m/s. Time of flight = 2 × 14.14 / 9.81 = 2.88 s. Range = 14.14 × 2.88 = 40.77 m. Max height = 14.14² / (2 × 9.81) = 10.19 m. Landing speed = 20.00 m/s (same as launch for level ground). These match the defaults above.

Related calculators

Free Fall Calculator: vertical drop with u = 0 and full gravity.
SUVAT Calculator: apply the full kinematic equations in one dimension.
Velocity Calculator: find constant velocity from distance and time.
Percent Error Calculator: compare theoretical range against a measured value.

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