The Elevator Problem

Episode 1 — MIT OCW 8.01x

Setup

2. Elevator Trip

A person is standing on an elevator initially at rest at the first floor of a high building. The elevator of height $h$then begins to ascend to the sixth floor, which is a known distance $L$ above the starting point. The elevator undergoes an unknown constant acceleration of magnitude $a$ for a given time interval $T$. Then the elevator moves at a constant velocity for a time interval $4T$. Finally the elevator brakes with an acceleration of magnitude $a$, (the same magnitude as the initial acceleration), for a time interval $T$ until stopping at the sixth floor.

(a) Make a sketch of the velocity $v(t)$ of the elevator as it travels to the sixth floor.

(b) Find the value of $a$, the magnitude of the acceleration, in terms of $L$ and $T$.

(c) If a = g, how fast would a person hit the ceiling?

Solving Part (b): Finding the Acceleration

The key insight is that distance equals the area under the v(t) curve. The velocity profile is a trapezoid — ramp up for time $T$, flat for $4T$, ramp down for $T$. The total area of the trapezoid must equal $L$.

What do we know?

Free Body Diagram

There are exactly two forces on you:

1. Gravity pulling down: $\vec{W} = -mg\,\hat{j}$
2. Normal force from the scale pushing up: $\vec{N} = N\,\hat{j}$

The scale reads the normal force $N$ — that's what "apparent weight" means.

Newton's Second Law

Applying $\vec{F}_{\text{net}} = m\vec{a}$ in the vertical direction:

Solving for the normal force:

where $a$ is the elevator's acceleration (positive upward).

What the Scale Reads

This single equation tells the whole story:

• Accelerating up ($a > 0$): $N > mg$ — you feel heavier
• Constant velocity ($a = 0$): $N = mg$ — normal weight
• Accelerating down ($a < 0$): $N < mg$ — you feel lighter
• Free fall ($a = -g$): $N = 0$ — weightlessness

The key insight: velocity doesn't matter. Only acceleration changes what the scale reads. You could be moving up at 100 m/s and the scale reads your normal weight — as long as the elevator isn't accelerating.

The Simulation

Play with the parameters and watch the forces in real time. Pay attention to the scale reading during each phase of motion.

Deeper: The Equivalence Principle

Einstein realized that no local experiment can distinguish between:

1. Standing in a gravitational field $g$
2. Accelerating upward at $g$ in empty space

If the elevator accelerates up at $g$, the scale reads $N = m(g + g) = 2mg$. But you'd get the exact same reading standing still on a planet with surface gravity $2g$.

This equivalence principle is the foundation of general relativity: gravity isn't a force at all — it's the curvature of spacetime.