During a 1987 skydive, parachutist Gregory Robertson saw that fellow skydiver, Debbie Williams, had suffered a mid-air collision with a third parachutist. Williams, rendered unconscious by the blow, was hurtling towards the ground at speeds in excess of 100 mph. At 13,500 ft, Robertson was well above Williams when he started to dive towards her. He reached a speed of about 200 mph, and catching up with Ms. Williams in mid-air, he went into a spread-eagle position to slow down and match her speed. As they both descended rapidly, and with seconds to spare, Robertson opened her chute and then his own, saving her life. He had expertly controlled his speed by changing his air resistance and therefore his drag coefficient. Even during a headfirst dive, Robertson reached an ultimate speed that he could not exceed, called his terminal velocity. Just what is terminal velocity and in particular a raindrop’s terminal velocity?
Any mass is attracted to the Earth by the pull of gravity. Gravity accelerates all objects towards the ground at a specific rate. Without any other forces present, the speed of an object in free fall will increase the farther or longer it falls. However, air friction or air resistance also exerts a force on an object (raindrop) that opposes the weight force of gravity. The air resistance and weight force on the droplet couple together to determine the terminal velocity for a given object.
In general the air resistance on an object depends upon several variables. First, it depends upon the shape of the object. Its shape determines the object’s drag coefficient: the more aerodynamic the shape, the less drag. Second, it depends upon the size of the object; specifically the cross-sectional area presented to the airflow (perpendicular to the direction of travel). And lastly, it depends upon the speed of the object. At low speeds the object's resistance is directly proportional to speed, and at higher speeds the object’s resistance is proportional to its speed squared. Most objects falling through the air would be considered to be moving at a higher speed, even though that speed might not be great compared to some velocities.
The speed at which an object falls increases until the upward force of air resistance equals the downward force of gravity, at which time the object reaches the terminal velocity. We know raindrops come in different sizes, so we need to consider an average size. Let us consider the average raindrop to have a radius of about 0.2 cm and a mass of about 0.034 grams. Aerodynamic engineers would give the rather round shape of a raindrop a drag coefficient of about 0.5. When all the parameters are considered the terminal velocity of a typical raindrop is calculated to be about 9 meters per second or 20 mph. A smaller raindrop of radius 0.15 cm has a terminal velocity of about 7 meters per second or 16 mph. In general, depending upon their size, raindrops fall between 15 and 25 miles per hour no matter how high they are when they begin their descent.
Not all falling objects have the low terminal velocity of raindrops. For example, you sometimes see people firing a gun into the air. Rifle bullets can exit the barrel with a speed of 2000 miles per hour. However, it returns to Earth at a terminal velocity of only about 200-mph due to air resistance. Although the projectile returns at only a fraction of its original speed, it is still enough to cause an injury.