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Gravitation problems in JEE ask you to find a satellite's speed or period from its height. In the early 1960s, getting those numbers right was a matter of life and death, and one of the people trusted to get them right was Katherine Johnson. This post covers her path to NASA, the problem her 1960 report solved, why an astronaut asked for her by name, a worked example using Friendship 7's real orbit, and what students can learn from how she worked.
Who was Katherine Johnson?
NASA's biography by Margot Lee Shetterly says she was born in White Sulphur Springs, West Virginia, in 1918, and was attending the high school on the campus of West Virginia State College by 13. She enrolled in the college at 18 and graduated with highest honours in 1937, with a mentor in maths professor W. W. Schieffelin Claytor. In 1939 she was one of three Black students chosen to integrate West Virginia University's graduate school.
After years of teaching, she joined the all-Black West Area Computing section at Langley in the summer of 1953. Within two weeks its head, Dorothy Vaughan, assigned her to the Flight Research Division. Our post on Dorothy Vaughan tells that side of the story.
What problem did her 1960 report solve?
After Sputnik in 1957, Langley's engineers turned to spaceflight. Johnson did trajectory analysis for Alan Shepard's Freedom 7 flight in May 1961. In 1960 she and engineer Ted Skopinski wrote Determination of Azimuth Angle at Burnout for Placing a Satellite Over a Selected Earth Position (NASA TN D-233).
The question it answers: when the rocket's engine cuts off, in which compass direction (azimuth) must the spacecraft be heading so that, after some orbits, it passes over a chosen spot on a rotating Earth? The report's summary says it relates the satellite's position in its orbital plane to latitude and longitude on a rotating Earth, includes corrections for Earth's oblateness (its slightly flattened shape), and checks the results against a computer program. NASA notes it was the first time a woman in the Flight Research Division was credited as an author of a research report.
Why did John Glenn ask her to check the computer?
For Glenn's orbital flight in 1962, NASA had programmed IBM computers with the orbital equations. The astronauts were wary of machines that were prone to hiccups and blackouts. According to the NASA biography, Glenn asked engineers to "get the girl", meaning Johnson, to run the same numbers through the same equations by hand on her desktop mechanical calculating machine. "If she says they're good," Johnson remembered him saying, "then I'm ready to go."
Worked example: how long did one Friendship 7 orbit take?
NASA's NSSDCA record for Mercury-Atlas 6 lists periapsis 159 km, apoapsis 265 km and a period of 88.50 minutes. Can you get that period from Kepler's third law?
Take Earth's radius R = 6371 km and GM = 3.986 × 10^14 m^3/s^2.
The orbit is almost circular (eccentricity 0.008), so use the semi-major axis: a = R + (159 + 265)/2 = 6371 + 212 = 6583 km = 6.583 × 10^6 m.
Kepler's third law: T = 2π √(a^3 / GM).
a^3 = (6.583 × 10^6)^3 ≈ 2.853 × 10^20 m^3.
a^3 / GM ≈ 2.853 × 10^20 / 3.986 × 10^14 ≈ 7.157 × 10^5 s^2.
√(7.157 × 10^5) ≈ 846.0 s, so T ≈ 2π × 846.0 ≈ 5316 s ≈ 88.6 minutes.
That matches the listed 88.50 minutes to within about 0.1%. Speed in a circular orbit: v = √(GM/a) ≈ 7.78 km/s, or about 28,000 km/h. The NSSDCA mission page says the spacecraft exceeded 28,000 km/hour.
Common mistakes:
Using the height (212 km) as the radius. Always measure from Earth's centre.
Mixing kilometres and metres inside the square root.
Forgetting the square root and getting a period in seconds squared.
For more practice, our guide to gravitation and satellite problems works through the standard JEE types.
What else did she work on?
The NASA biography says that when asked about her greatest contribution, Johnson would talk about the calculations that helped sync Apollo's Lunar Module with the orbiting Command and Service Module. She also worked on the Space Shuttle and on the Earth Resources Technology Satellite (later Landsat), authored or co-authored 26 research reports, and retired in 1986 after 33 years at Langley. In 2015, at 97, she received the Presidential Medal of Freedom. She died on 24 February 2020. The Hidden Figures Congressional Gold Medal Act of 2019 awarded her a Congressional Gold Medal.
What can a student take from her way of working?
Check by a different route. Johnson's reputation rested on independent verification.
Know the geometry, not just the formula. Her report is about angles on a rotating, slightly flattened sphere, the kind of picture you should sketch before any orbit problem.
Put your name on your work. Being credited as an author mattered, and it opened doors.
Orbits are easier to understand when you can watch one change. The Fermi Simulation Lab has interactive simulations for ideas that are easier to see than to read, so you can see how speed and radius trade off in a stable orbit.
Next step: repeat the worked example for a satellite 400 km up and check your period against the 92 to 93 minutes you should get.
