Projectile Motion
Calculate range, maximum height, and time of flight for a projectile.
Formula
R = v²sin(2θ)/g
Range depends on initial velocity squared and the sine of twice the launch angle.
Worked Examples
Example 1: Cannonball
Problem:100m/s at 45°
Solution:100^2 / 9.81
Result:Range: 1019m
Frequently Asked Questions
What is projectile motion?
Projectile motion describes an object launched into the air and then acted on by gravity alone, with air resistance ignored. The key idea is that the horizontal and vertical components are independent: horizontally the object moves at constant velocity, while vertically it accelerates downward at 9.81 m/s2. Combining a constant horizontal motion with a uniformly accelerated vertical motion produces the familiar parabolic arc.
Why is 45 degrees the optimal launch angle for range?
Range is given by R = v2 sin(2θ) / g, so range is maximised when sin(2θ) reaches its maximum of 1, which happens when 2θ = 90 degrees and therefore θ = 45 degrees. This holds only when launch and landing are at the same height and drag is ignored. Launching from an elevated position lowers the optimal angle, and real projectiles subject to air resistance peak nearer 35 to 40 degrees.
How do you calculate time of flight?
For level ground, time of flight is t = 2 v sin(θ) / g, twice the time taken to reach the apex. A ball launched at 20 m/s and 30 degrees has a vertical component of 20 x sin(30) = 10 m/s, so it takes 10 / 9.81 = 1.02 s to reach the top and 2.04 s in total. If the landing point is lower than the launch point you must solve the full quadratic instead.
Does the mass of the projectile matter?
Not in a vacuum. Gravitational force is proportional to mass, but so is the inertia resisting that force, and the two cancel in Newton second law to leave an acceleration of g regardless of mass. This is Galileo insight, confirmed by David Scott dropping a hammer and a feather on the Moon in 1971. In air, mass does matter indirectly, because a heavier object of the same shape is decelerated less by the same drag force.
Why is the trajectory a parabola?
Horizontal position grows linearly with time, x = v cos(θ) t, while vertical position follows y = v sin(θ) t minus half g t squared. Solving the first equation for t and substituting into the second leaves y as a quadratic function of x, and every quadratic in one variable plots as a parabola. The parabola is therefore a direct consequence of constant horizontal velocity combined with constant vertical acceleration.
How much does air resistance change the answer?
Considerably for light or fast objects. Drag scales roughly with the square of speed, so it removes energy fastest early in the flight and makes the descending half of the path steeper than the ascending half, breaking the parabolic symmetry. A well-struck golf ball that would travel about 400 m in a vacuum carries roughly 250 m in air, and a badminton shuttlecock is affected so strongly its path is almost triangular.
What is the vertical velocity at the highest point?
Zero. The projectile is still moving horizontally at v cos(θ), but its vertical velocity has been reduced to nothing by gravity before reversing sign on the way down. This is why the apex is the correct place to evaluate maximum height: setting the vertical velocity to zero in v squared = u squared minus 2 g h gives h = v squared sin squared(θ) / 2g directly.
What are the standard projectile motion formulas?
For launch speed v at angle θ from level ground: range R = v2 sin(2θ) / g, maximum height H = v2 sin2(θ) / 2g, and time of flight T = 2 v sin(θ) / g. Position at any time t is x = v cos(θ) t and y = v sin(θ) t minus 0.5 g t2. All three summary formulas assume launch and landing at the same elevation and no air resistance.
Where is projectile motion used in practice?
Ballistics and artillery firing tables are the classic application, along with sports analysis for golf drives, basketball shot arcs and long jump take-off angles. Engineers use it to design fountain nozzles, irrigation sprinkler coverage and conveyor discharge chutes. Film and game physics engines integrate the same equations frame by frame, adding drag terms when realism matters more than the closed-form solution.
What are Newton's three laws of motion?
Newton's first law states that an object at rest stays at rest and an object in motion stays in motion unless acted on by an external force. The second law relates force, mass, and acceleration: F = ma. The third law states that for every action there is an equal and opposite reaction.