* The ISS is visible to the naked eye, so students can measure the motion, then analyze it as an exercise in angular velocity and orbital dynamics. Measure the angular velocity and use this to calculate the speed of the space station in orbit. Equipment required is completely free!*

###### Basic Idea of the Experiment

Fingers held at arm’s length are calibrated to provide an angular measure. As the space station crosses each finger on the hand of an up-stretched arm, record the time on a stopwatch or cellphone stopwatch App. This gives the angular velocity. Combine this with the distance up to the space station to get the ISS velocity.

###### Materials

- Ruler to measure width of hand and distance from eye to hand.
- Information about the ISS schedule of crossings in your locale. This can be found at http://spotthestation.nasa.gov/sightings/#.VTHyIfCGOi8

###### Experimental procedure:

- Find out a good date and time and spot the ISS. The above website – Spot the Station – shows
- dates and times
- starting direction and elevation where the shuttle first becomes visible
- the ending of visibility direction and elevation
- the maximum elevation for the traversal.

The ISS will be visible only in the morning and the evening, because for most of the night, the Earth casts its shadow over the shuttle orbit. The shuttle travels from generally West to generally East; therefore it is very brightly illuminated soon after it rises in the morning, and a little less brilliant when it rises in the evening, due to the relative positions of the sun, the ISS, and the observer.

- Take measurements when the ISS is near its highest elevation. Hold the hand still above your head with the palm and fingers bent horizontal. As the ISS reaches the pinky finger, call out the start. Then call out as it crosses the three narrow spaces between fingers, as well as when it emerges past the index finger.
**Hint:**holding the hand absolutely still is difficult. It may help if there is a star or tree branch above the hand to help you steady your hand. - Measure the average angle in radians subtended by the four fingers. This is just the average width of the fingers divided by the distance from the hand to the eye.

###### Calculations

Let the finger width be w, and the distance from the eye to the finger be r. Then the angular displacement of any object whose image crosses the finger is Δθ = w/r. Let Δt be the time to cross over a finger. Then calculate the angular velocity as ω = Δθ/Δt = w/(rΔt).

To find the velocity of the ISS, multiply the angular velocity by the distance R from the eye to the ISS: v = ωR. A slight complication is that the distance from the observer to the ISS depends on the elevation angle. If the ISS passes directly overhead, so maximum elevation = 90ᵒ, then R is just the typical distance between the Earth and the space station, R = 249 +_ 5 mi = km. If the maximum elevation angle of the ISS is θ, then R = 249/sin θ.

**Example:**

On the morning of March 30, 2015, in Phoenix, AZ, USA, the table in the Spot the Station website shows that the ISS first became visible 10ᵒ above the horizon, reached a maximum elevation of 77ᵒ, and travelled for ≈ 6 minutes before disappearing at an elevation of 12ᵒ.

Holding my hand with fingers straight but slightly apart, so I could see narrowly between my fingers, *my* hand-width is 8.0 cm. Hence each finger averages 2.0 cm width. The average time to cross a finger was 3.0 s.

When I hold my bent hand above my head, the fingers are 33 cm above my eyes. Therefore the angular velocity of the space station appeared to me as ω = (2.0 / 33 * 3.0) = 0.0202 radians/sec. If the ISS had passed directly overhead, the velocity would just be ω*R, with R = 249 miles. But since the elevation was 77ᵒ instead of 90ᵒ, the perpendicular distance between the object and me (the observer) was actually the hypotenuse of a triangle whose height was 249 miles and whose angle was 77ᵒ. Therefore R = 249 / sin 77ᵒ = 256 miles. Converting to kilometers, R = 256 mi *1.61 km/mi = 411 km, and v = 0.0202 * 411 = 8.3 km/s. This is in reasonable agreement with look-up values – 7.6-7.7 km/s.

###### Discussion

All low orbits — including the space station– move at about 5 miles (or 8 km) per second. Since the space station is about 250 miles above the Earth, the angular velocity of motion past an earthbound observer is about 5 /250 = .02 radians per second.

It is exciting to realize that while you are viewing the ISS for 6 minutes, it is moving across the country a distance of 300 s x 5 = 1500 miles or 2400 km. It is exciting to think that when the ISS is just going out of your view, it is passing over some far-away place that is 750 miles distant!