Orbital Speed: Using the Equation

Explained

Orbital speed: one equation, three ways to be asked about it

Orbital speed uses v equals two pi r divided by T. It looks unfamiliar, but it is only speed equals distance divided by time in disguise. Two pi r is the circumference of the orbit, which is the distance travelled in one lap, and T is the time that lap takes.

What each symbol needs to be

• v is the orbital speed in metres per second.
• r is the orbital radius in metres, measured from the centre of the object being orbited to the centre of the orbiting object.
• T is the orbital period in seconds, meaning the time for one complete orbit.

The radius is measured centre to centre, not from the surface. A satellite orbiting 300 km above the Earth has an orbital radius of the Earth's radius plus 300 km, and questions frequently supply the height rather than the radius to see whether you notice.

Units are where the marks go

Orbital periods are almost never given in seconds. They come as days, hours or years, and every one of them must be converted before the equation is used.

A day is 86,400 seconds. A year is roughly 3.15 times ten to the seven seconds. Radii arrive in kilometres more often than metres, so multiply by 1,000.

Do the conversions first, write them down, and then substitute. An answer that is out by a factor of 1,000 or 86,400 is almost always a conversion that was skipped rather than a misunderstanding of the physics.

Rearranging it

The equation is asked in all three directions, so practise all three.

• For radius: r equals v times T divided by two pi.
• For period: T equals two pi r divided by v.

If rearranging under pressure feels risky, substitute the numbers you have into the original equation first and then solve for the unknown. It takes slightly longer and goes wrong far less often.

What the equation tells you about orbits

For objects orbiting the same body, a larger orbital radius means a longer period and a slower orbital speed. Mercury is closer to the Sun than Neptune, and it orbits both faster and in far less time.

If a question gives you data for several planets or moons and asks you to comment, that relationship is usually the point being tested rather than the arithmetic.