How can I calculate vertical speed?

ChatGPT says:

What you’re describing is partially correct in theory, but it runs into serious practical problems when you try to do it with a phone’s accelerometer.

Let’s break it down.

1. The theory

You’re using two standard kinematic equations:

1. Distance

In an ideal world:

• You’d take acceleration data from the accelerometer.
• Integrate once to get velocity.
• Integrate again to get displacement.

That’s exactly what’s in your equations.

2. Why this breaks in real life

Phone accelerometers are noisy and measure total acceleration, which includes:

• Gravity (~9.81 m/s²) — which is huge compared to small motion changes.
• Sensor bias (a constant tiny offset error).
• Random noise.

When you integrate noisy acceleration:

• Even a 0.01 m/s² bias turns into a large velocity error in seconds.
• A small velocity error turns into a massive position error when integrated again.

That’s why:

• In practice, purely integrating accelerometer data without correction drifts like crazy.
• After just a few seconds, your “distance” can be completely wrong — sometimes by meters.

3. How variometer apps make it work

Variometers (used in paragliding, skydiving, etc.) do detect vertical speed from accelerometers, but:

• They don’t rely on just raw acceleration integration.
• They use sensor fusion with the barometer (altimeter) to correct drift.
• The barometer provides an absolute vertical position (pressure → altitude), while the accelerometer fills in short-term responsiveness.
• Some also use GPS as a correction source (slow, but drift-free over time).

Without a barometer, your vertical velocity estimate will drift almost immediately.

Bottom line

• Possible in principle: Yes, your physics equations are fine.
• Accurate with a phone accelerometer alone: No, because of noise, bias, and gravity removal issues.
• How apps do it: They combine an accelerometer, gyroscope, and barometer (and sometimes GPS).
• If you want to make a working vertical velocity sensor, Start by learning how to remove gravity via gyroscope data, then combine accelerometer with barometer readings to prevent drift.

Typically air pressure was used to do such calculations, not acceleration. The variometer app you mentioned might be using barometric sensors to calculates vertical velocity?

In free fall, acceleration is actually "0", this is likely the value the accelerometer reports.