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Once a ball leaves your hand, nothing pushes it forward any more. It keeps moving sideways at a steady speed while gravity pulls it down, a little more each second. Put those two simple ideas together and you get a smooth curve called a parabola.
Two motions at once
- 1Split the speedA launch speed v at angle θ gives a sideways part vx = v cos θ and an upward part vy = v sin θ.
- 2Sideways: no forcevx never changes (we ignore air). Distance sideways = vx × time.
- 3Up and down: gravityvy drops by g = 9.81 m/s every second. Up for a while, then down.
- 4Join themUse the same time for both. The ball lands when its height is back to zero.
This is the trick behind every projectile problem in school physics. The calculator below does the same four steps.
Five angles, one speed
Take a ball thrown at 20 m/s from the ground on Earth. Only the angle changes. Notice how 15° and 75° land at the same spot, and so do 30° and 60°, but the high throws spend much longer in the air.
| Angle | Range | Peak height | Time in air |
|---|---|---|---|
| 15° | 20.4 m | 1.4 m | 1.06 s |
| 30° | 35.3 m | 5.1 m | 2.04 s |
| 45° | 40.8 m | 10.2 m | 2.88 s |
| 60° | 35.3 m | 15.3 m | 3.53 s |
| 75° | 20.4 m | 19.0 m | 3.94 s |
Range = v² sin(2θ) ÷ g and peak = v² sin²θ ÷ 2g with v = 20 and g = 9.81. Rounded to one decimal.
The bars rise to a peak at 45° and fall again in a mirror image. The mirror pairs always add up to 90°.
The 30° and 60° paths touch the ground at nearly the same place (about 35 m), but the 60° ball climbs three times higher. The 45° path travels farthest. Values after landing are shown as zero.
Why 45 degrees wins
A steeper throw stays in the air longer but spends its speed going up. A flatter throw keeps more sideways speed but lands quickly. The product of the two, sideways speed times time, is biggest when they are balanced, and that balance point is exactly 45° on level ground.
- Most speed goes sideways
- Hits the ground in about 1 second
- Range: 20.4 m
- Speed split equally
- About 2.9 seconds in the air
- Range: 40.8 m, double the 15° throw
A 75° throw has the mirror problem: lots of time, hardly any sideways speed.
Speed matters more than angle
Because the range formula has v², small speed gains give big distance gains. At 45° on Earth, 10 m/s gets you about 10 m, but 40 m/s gets you about 163 m, sixteen times more for four times the speed.
Double the speed, four times the range. This is why a small increase in a cricket bowler's or a golfer's speed matters so much.
Same throw, different worlds
Gravity sets how fast the ball falls back. With the same 20 m/s, 45° throw, the range is simply v² ÷ g, so a weaker pull means a much longer flight.
On the Moon the same throw lasts about 17.5 seconds and travels nearly 250 metres. These are ideal numbers with no air; Jupiter has no ground to stand on, so treat it as a thought experiment.
When 45 degrees is not the best
The 45° rule needs the ball to start and land at the same height. Throw from a rooftop, a cliff or a bowler's raised arm and the best angle drops below 45°, because the ball has extra time to fall. A shot-putter releases at about 2 m above ground and the ideal angle is a little lower than 45° (for 20 m/s it is about 43.7°). In real sport, air drag, spin and a body that cannot throw as hard at steep angles pull the best angle lower still: long jumpers take off near 20°, and good javelin throws leave at around 30° to 35°.
The ideal formula is a starting point. Real throwing sports tune the angle by trial and by feel.
Using the calculator well
- Use consistent units. Metres and seconds, or feet and seconds, never a mix.
- Check the starting height. Zero means level ground; a positive number means a raised launch.
- Try the mirror angle. Enter 30° then 60° and watch the range stay the same while time and height change.
- Treat results as ideal. Wind, spin and drag will shorten real throws, especially fast ones.
Questions people ask
What is projectile motion?
It is the motion of an object thrown or launched into the air that moves only under gravity. Its path is a parabola because the sideways speed is constant while gravity changes the vertical speed.
Why does 45 degrees give the longest range?
On level ground the range is v² sin(2θ) ÷ g. The sine is largest, equal to 1, when 2θ is 90 degrees, so θ is 45 degrees.
What is the formula for time of flight?
For a launch from the ground, time in the air is 2 × v × sin θ ÷ g. At 20 m/s and 45° that is about 2.88 seconds on Earth.
Do two different angles give the same range?
Yes. Any two angles that add up to 90 degrees, such as 30° and 60°, give the same range on level ground. The higher one stays in the air longer.
How does air resistance change things?
Drag slows the object, shortens the range and lowers the best launch angle. The calculator assumes no air, so real throws, especially fast and light ones, fall shorter.
What is the acceleration due to gravity?
On Earth it is about 9.81 m/s² near the surface, often rounded to 9.8 or 10. It is about 3.71 on Mars and 1.62 on the Moon.