How Far Can Bacteria Coast?

Random Walks in Biology, by Howard Berg.

In last week’s blog post, I told you about the recent death of Howard Berg, author of Random Walks in Biology. This week, I present a new homework problem based on a topic from Berg’s book. When discussing the Reynolds number, a dimensionless number from fluid dynamics that is small when viscosity dominates inertia, Berg writes

The Reynolds number of the fish is very large, that of the bacterium is very small. The fish propels itself by accelerating water, the bacterium by using viscous shear. The fish knows a great deal about inertia, the bacterium knows nothing. In short, the two live in very different hydrodynamic worlds.

To make this point clear, it is instructive to compute the distance that the bacterium can coast when it stops swimming.

Here is the new homework problem, which asks the student to compute the distance the bacterium can coast.

Section 1.20

Problem 54. When a bacterium stops swimming, it will coast to a stop. Let us calculate how long this coasting takes, and how far it will go.

(a) Write a differential equation governing the speed,v, of the bacterium. Use Newton’s second law with the force given by Stokes law. Be careful about minus signs.

(b) Solve this differential equation to determine the speed as a function of time.

(c) Write the time constant,τ, governing the decay of the speed in terms of the bacterium’s mass, m, its radius, a, and the fluid viscosity, η.

(d) Calculate the mass of the bacterium assuming it has the density of water and it is a sphere with a radius of one micron.

(e) Calculate the time constant of the decay of the speed, for swimming in water having a viscosity of 0.001 Pa s.

(f) Integrate the speed over time to determine how far the bacterium will coast, assuming its initial speed is 20 microns per second.

I won’t solve all the intermediate steps for you; after all, it’s your homework problem. However, below is what Berg has to say about the final result.

A cell moving at an initial velocity of 2 × 10^–3 cm/sec coasts 4 × 10^–10 cm = 0.04 , a distance small compared with the diameter of a hydrogen atom! Note that the bacterium is still subject to Brownian movement, so it does not actually stop. The drift goes to zero, not the diffusion.

Berg didn’t calculate the deceleration of the bacterium. If the speed drops from 20 microns per second to zero in one time constant, I calculate the acceleration to be be about 91 m/s^2, or nearly 10g. This is similar to the maximum allowed acceleration of a plane flying in the Red Bull Air Race. That poor bacterium.

Originally published at




Professor of Physics at Oakland University and coauthor of the textbook Intermediate Physics for Medicine and Biology.

Love podcasts or audiobooks? Learn on the go with our new app.

Recommended from Medium

“Ion Implantation ” Science-Research, December 2021, Week 2 — summary from Astrophysics Data…

Quantum Computing on the cover of Scientific American Brazil, June 2022!

Why do we like to eat the same thing every morning?

“Metal Complexes” Science-Research, December 2021, Week 4 — summary from NASA Technology Transfer…

“Quantum entanglement” Science-Research, October 2021, Week 4 — summary from Arxiv

“X-ray” Science-Research, March 2022, Week 1 — summary from DOAJ, Europe PMC, Springer Nature…

Jacqueline K. Barton, Chemist & Pioneer in the Study of DNA Helix Properties

“Embryonic Stem Cells” Science-Research, November 2021 — summary from DOAJ, PubMed and Springer…

Get the Medium app

A button that says 'Download on the App Store', and if clicked it will lead you to the iOS App store
A button that says 'Get it on, Google Play', and if clicked it will lead you to the Google Play store
Brad Roth

Brad Roth

Professor of Physics at Oakland University and coauthor of the textbook Intermediate Physics for Medicine and Biology.

More from Medium

Posing questions to creators and regulators of AI can help shape the safety of digital spaces for…

Future of IoT

A Comparison Between Three Genome Editing Techniques: ZFNs, TALENs, and CRISPR

What’s the real purpose of the Teslabot?