Projectile motion
Launch a projectile and watch how the horizontal velocity stays constant while gravity first slows down and then speeds up the vertical motion.
The equations of projectile motion
Projectile motion is the combination of two independent motions. Horizontally no force acts (if we neglect air resistance), so the projectile moves at constant velocity. Vertically, gravity acts and produces uniformly accelerated motion. If we launch with speed v₀, angle θ and from a height h₀:
x(t) = v₀·cos θ · t
y(t) = h₀ + v₀·sin θ · t − ½·g·t²
Eliminating time between the two equations gives a parabola, hence the name. Turn on the vectors in the simulator: the horizontal arrow (vx) never changes, while the vertical one (vy) shrinks on the way up, vanishes at the highest point and grows downwards as the projectile falls.
Maximum height and range
At the highest point the vertical velocity is zero, which happens at t = v₀·sin θ / g. Substituting:
hmax = h₀ + (v₀·sin θ)² / (2g)
When launched from the ground (h₀ = 0), the range is R = v₀²·sin(2θ) / g. The sine of 2θ is largest when θ = 45°, and since sin(2θ) = sin(180° − 2θ), two complementary angles (for example 30° and 60°) land at exactly the same spot: the higher one takes longer and climbs higher, but covers the same distance. The dashed trajectory in the simulator shows that complementary angle.
What if you launch from a height?
When h₀ > 0 the optimal angle is no longer 45° but somewhat lower, because the projectile has more time to fall and it pays to use it moving horizontally. The angle for maximum range is θ* = arctan(v₀ / √(v₀² + 2gh₀)). Try raising the initial height and watch that value drop.
The simulator ignores air resistance. In reality, a ball or a bullet travels a good deal less far and the optimal angle is below 45°.
Frequently asked questions
What launch angle gives the maximum range?
If the projectile lands at the same height it was launched from and air resistance is ignored, the maximum range is reached at 45°, because sin(2θ) equals 1. When launched from a height, the best angle is somewhat below 45°. Air drag lowers it too: in sports such as shot put it is usually between 35° and 42°.
How do you calculate the range of a projectile?
Launching from the ground, the range is R = v₀² · sin(2θ) / g. For example, a ball thrown at 20 m/s and 45° on Earth (g = 9.81 m/s²) travels about 40.8 m. The time of flight is t = 2 · v₀ · sin θ / g, about 2.9 s in that example.
Why do two different angles give the same range?
Because sin(2θ) has the same value for two complementary angles, θ and 90° − θ. Launching at 30° or at 60° with the same speed lands at the same spot, but at 60° the projectile climbs much higher and stays in the air longer. Only at 45° do both angles coincide.
How much farther would an object go on the Moon?
Range is inversely proportional to gravity. On the Moon g is 1.62 m/s², about 6 times less than on Earth, so with the same speed and angle the projectile travels about 6 times farther and climbs about 6 times higher. On Jupiter, at 24.8 m/s², it would cover less than half the distance.