Air resistance and terminal velocity

Thrown at 30 m/s and 45°, a ball would land 92 m away in a vacuum; with air pushing back, our model ball lands at about 60 m.

Air pushes back

Put your hand out of a car window and you feel air resistance, also called drag. It always points against the motion, and it grows quickly with speed. For a ball, a bike or a car moving at ordinary speeds through air, OpenStax College Physics gives the drag force as F_D = ½ C ρ A v². Here ρ is the density of the air, A is the area facing the flow, C is a drag coefficient that depends on the shape, and v is the speed. Wind tunnel tests give C about 0.45 for a sphere, 0.28 for a streamlined family car and about 1.0 for a skydiver lying flat.

The v² is the important part. Double the speed and the drag becomes four times larger. A slow ball hardly notices the air; a fast one is pushed hard. That is why air resistance matters little for a gently tossed beanbag and a great deal for a hit baseball or a fast car, where OpenStax notes that more than half the engine’s power at highway speed goes into pushing air aside.

A lopsided path

Launch the model ball in the lab at 30 m/s and 45° with air resistance on. Without air it would fly 91.7 m. With air it lands at about 60.0 m, a third shorter. The path is no longer a symmetric parabola. It climbs at a steep angle and then drops more steeply than it rose, because the ball has lost much of its horizontal speed by the time it comes down. The highest point also moves closer to the landing spot.

Now switch to Mars. Mars has air, but NASA puts its surface density at about 0.016 kg/m³, compared with about 1.2 kg/m³ on Earth, so the drag is roughly 75 times weaker. The kept paths for Mars with and without air almost overlap. On the Moon there is essentially no air at all, and the lab draws exactly the same path whether the switch is on or off.

Terminal velocity

Think about a skydiver jumping from a plane. At first the speed is low, so drag is small and the skydiver accelerates at almost g. As the speed rises, drag grows, and the net downward force shrinks. Eventually drag equals weight. The forces balance, the acceleration drops to zero and the speed stops increasing. This top speed is the terminal velocity. Setting ½ C ρ A v² equal to m g gives v_t = √(2 m g ÷ ρ C A).

OpenStax works two examples. A 75 kg skydiver diving head first, with a frontal area of about 0.18 m² and C ≈ 0.70, reaches about 98 m/s, roughly 350 km/h. An 85 kg skydiver spread flat, with about 0.70 m² of area and C ≈ 1.0, reaches only about 44 m/s. Our model ball in the lab has a terminal velocity of about 35 m/s, which is why fast launches lose so much range. Opening a parachute increases A and C enormously, and the terminal velocity drops to a speed you can land at.