Calculator: Throw Distance at Angle

Calculate the distance of an object thrown at a certain angle, without air resistance. The air resistance is neglectable at compact, heavy objects and a velocity reached at normal throwing, it only slightly decreases the distance. So the distance depends on launch speed and the trajectory angle. A 45 degrees angle is ideal if, as assumed here, launch and landing point have the same height. The trajectory is an upside down parabola. The formula for the distance is d = v² * sin(2α) / a. The higher the launch speed, the greater the air resistance and the more this calculation overestimates the throwing distance.
Launch velocity or thrown distance can be calculated from the other three values.

Example: a stone, thrown with 70 kilometers per hour, can fly up to 38.5 meters. But thrown at a 30 degrees angle, it flies only 33 meters, the same in an angle of 60 degrees.
70 km/h is already a high throwing speed, and air resistance plays a small role. If you want to take air resistance into account, you need more information about the stone and the air, and you also have to integrate the calculations numerically, which is mathematically quite demanding. If you assume a round stone with a diameter of 5 centimeters, a material density of 2700 kilograms per cubic meter, and an average air density at sea level, you get a throwing distance of 35.2 meters, or about ten percent less. At higher speeds, air resistance increases and so the throwing distance decreases more. For example, if you are at an altitude of 1000 meters above sea level, the air density drops from about 1.225 to 1.1 kilograms per cubic meter, increasing the throwing distance by about a quarter of a meter.