Connect circular speed, period and position

A circular orbit needs a particular speed

Choose radius 2 Earth radii and elapsed model time zero. The spacecraft begins on the positive horizontal axis, moving counterclockwise. Predict its next direction before increasing elapsed time. It should move upward and curve around Earth, not travel directly toward the center along the acceleration arrow.

In circular motion, the required inward acceleration has magnitude v²/r, where v is speed and r is radius in consistent units. Gravity provides μ/r². Equating them gives v²/r = μ/r², and therefore v = √(μ/r). This is the speed appropriate to the chosen circle, not a command that a real spacecraft automatically follows after any disturbance.

At r = 12742 km and μ = 398600 km³/s², v = √(398600/12742), about 5.5931 km/s. The units inside the square root are km²/s², so the result is kilometers per second. The parameters are rounded from the NASA Earth fact sheet.

Connect distance, speed and period

A circle’s circumference is 2πr. Dividing this distance by constant speed gives the period T = 2πr/v, the time for one revolution. Substituting the circular-speed expression gives T = 2π√(r³/μ). With kilometers and seconds in μ, this formula returns seconds. Divide by 60 to compare with the time slider’s minutes.

For the default circle, the period is about 238.5703 minutes. Thirty minutes is about 0.12575 of one revolution. Multiplying by 360° gives approximately 45.2697° of accumulated motion. This calculation connects a point on the diagram with elapsed time instead of treating its position as decorative animation.

On paper, draw the default circle and mark the predicted 30-minute location. Then compare the model. Your drawing need not measure the angle to four decimal places; explain the scale of the prediction and keep rounding consistent with a hand calculation.

Distinguish orientation from accumulated motion

Orientation repeats after each full revolution. Accumulated angle does not. If the spacecraft completes one revolution and another quarter, its accumulated angle is 450°, but its orientation is 90°. A wrapped angle removes complete multiples of 360° to locate the current direction.

This difference matters at the smaller radius 1.1R, whose period is about 97.3107 minutes. At 240 minutes it has completed more than two revolutions. A single position marker cannot show how many times it has passed that location. Inspect elapsed time and total angle alongside wrapped orientation rather than inferring travel history from one point.

For a paper exercise, use a hypothetical circular period of 80 minutes. Calculate accumulated angle and orientation after 100 minutes: 450° and 90°. This arithmetic example is independent of the slider and checks whether you understand wrapping.

Read vectors at the current position

The position coordinates are r cos θ and r sin θ, with θ measured counterclockwise from the positive horizontal axis. Cosine and sine give the horizontal and vertical projections of a radius. At 0° the position is (r, 0); at 90° it is (0, r).

The velocity is tangent and acceleration is inward at every position. Equal speed does not make the velocity vector constant. NASA’s gravity and mechanics explanation connects orbital motion with gravity; this lab narrows that physical idea to an analytically enforced circle in a nonrotating plane.

No Earth rotation or geographic surface coordinates are calculated. The upper point of the drawing is not automatically above the North Pole, and crossing the positive axis does not imply passing over the same city. The reference circle at 2R shares elapsed model time and starting direction, making comparisons controlled but constructed.

Transfer beyond the displayed time

For a paper-only orbit with a 120-minute period, find accumulated angle and wrapped orientation after 330 minutes. Does returning to a previous orientation imply zero acceleration?

Check the circular-orbits transfer

330/120 = 2.75 revolutions, giving 990° accumulated angle and 270° orientation. Inward acceleration persists throughout the circular path; repeated orientation does not remove gravity or make velocity unchanging.