Apparent Weight, Elevators, and Terminal Velocity
Apparent weight is the normal force that a supporting surface exerts on an object, which equals the weight mg only when there is no vertical acceleration; in an elevator accelerating upward at a, the apparent weight is m(g + a), and a falling object reaches terminal velocity when drag equals its weight. In brief: you feel heavier when the elevator starts moving up or stops while going down, lighter when it starts down or stops going up, and weightless in free fall. Terminal velocity depends on the mass, the shape and the density of the fluid: vt = √(2mg ÷ ρCdA).
When an elevator starts to move, we feel a change in the force of the floor on our feet. This guide explains apparent weight in accelerating frames, free fall and weightlessness, and the approach to terminal velocity, compares the situations in a table, works through numerical examples and corrects common mistakes.
What is apparent weight?
The weight of an object is the gravitational force mg. What a scale or our body senses is the normal force N exerted by the surface on which we stand. When there is no vertical acceleration, N = mg. When the surface accelerates vertically, N is different: Newton’s second law for a person standing in an elevator gives N − mg = ma (taking upward as positive), so the apparent weight is N = m(g + a). The sensation of weight is the contact force, which is why a person in free fall (a = −g, N = 0) feels weightless, although gravity still acts on them.
What happens in an elevator?
- Accelerating upward (starting to go up, or slowing while going down): a is positive, N > mg, and you feel heavier.
- Constant velocity (up or down): a = 0, N = mg, and you feel normal.
- Accelerating downward (starting to go down, or slowing while going up): a is negative, N < mg, and you feel lighter.
- Free fall (a cable break, ignoring safety devices): a = −g, N = 0, weightlessness.
The same principle explains the feeling of heaviness at the bottom of a roller coaster dip, and the weightlessness of astronauts in orbit, who are in continuous free fall around the Earth, and of passengers in parabolic flights, which are used to train astronauts.
What is terminal velocity?
When an object falls through a fluid, drag grows with speed. When the drag equals the weight, the net force is zero and the object falls at a constant terminal velocity. At first the acceleration is g, and it decreases as the speed rises, approaching zero. For quadratic drag Fd = ½ρCdAv2, the terminal velocity is vt = √(2mg ÷ ρCdA). It increases with mass (for the same shape) and decreases with area and drag coefficient. A skydiver reaches about 55 m/s (200 km/h) when falling flat, about 90 m/s in a head-down position, and about 5 to 6 m/s with the parachute open, as the area is much larger. Rain drops reach their terminal speed within a short distance, between about 2 m/s for drizzle and 9 m/s for large drops.
Situations compared
| Motion of the elevator | Acceleration | Apparent weight | Sensation |
|---|---|---|---|
| Starting upward | a upward | m(g + a) | Heavier |
| Constant speed | 0 | mg | Normal |
| Starting downward | a downward | m(g − a) | Lighter |
| Free fall | g downward | 0 | Weightless |
Worked example: elevator and parachute
- Elevator accelerating up. A person of mass 70 kg stands on a scale in an elevator that accelerates upward at 2.0 m/s2. N = 70 × (9.8 + 2.0) = 826 N, compared with the real weight of 686 N. The scale reads 826 ÷ 9.8 = 84 kg.
- Accelerating down. If the acceleration is 2.0 m/s2 downward: N = 70 × (9.8 − 2.0) = 546 N, equivalent to 56 kg.
- Constant velocity. At a constant speed, the scale reads the true 686 N (70 kg).
- Cable tension. The elevator has a total mass of 1,000 kg (with passengers) and accelerates upward at 1.5 m/s2. The tension T − mg = ma, so T = 1,000 × (9.8 + 1.5) = 11,300 N.
- Parachutist. A parachutist of mass 80 kg with a canopy area of 25 m2, Cd = 1.5, in air of density 1.2 kg/m3. vt = √(2 × 80 × 9.8 ÷ (1.2 × 1.5 × 25)) = √(1,568 ÷ 45) = √34.8 = 5.9 m/s.
- Heavier object of the same shape. If the parachutist’s mass were 120 kg, vt = 5.9 × √(120 ÷ 80) = 5.9 × 1.22 = 7.2 m/s.
How were these ideas developed?
Einstein described in 1907 how a person in a freely falling elevator would feel weightless, and he called this his “happiest thought”, leading to the equivalence principle of general relativity: the effects of gravity and acceleration cannot be distinguished locally. George Stokes derived the drag on a sphere in a viscous fluid in 1851, and research on terminal velocity of falling bodies was important for parachutes in the early 20th century.
Which mistakes are common?
Mistake 1: Saying that the weight changes in an elevator
The gravitational force stays the same. The apparent weight (the normal force) changes.
Mistake 2: Thinking that astronauts in orbit escape gravity
Gravity is about 90 percent of its surface value at the altitude of the space station. They are in free fall.
Mistake 3: Thinking that the object stops accelerating at the terminal velocity because forces vanish
The forces are still large, but they balance each other.
Mistake 4: Assuming that terminal velocity is the same for all objects
It depends on the mass, area, shape and the density of the fluid.