Thursday, December 22, 2011

Measuring Height with a Video Camera

There are many pages on the internet with a story about Niels Bohr, Noble laureate for physic, from Denmark, when he was in college (the University of Copenhagen). One writing is on this wikipedia page.

The purpose of the story is to teach students to think creatively, in order to get a new idea, which has never occurred before, or out of the box.

One day, a teacher gave a question to the students, one of them was Niels Bohr, on how to measure the height of a skyscraper with a barometer. Some answers were:
• measure the shadow of the building at about 11 AM to 1 PM, when shadow has short length, and then compare with the shadow of an object with known height
• tie the barometer to a rope and then let it down, the length of rope is equal to the height of the building.
• dropping barometer and record the time so the height can be calculated

I am interested to discuss the final answer that is by recording the time length of a falling object. In this case I do not intend to drop an expensive barometer, but it can be other objects that are cheaper and not dangerous such as: tennis ball, basket ball, watermelon, etc.. Tennis balls may be the best choice because of its striking color and big enough to be visible from a height when it reaches the ground, tennis balls are also not dangerous if it falls on men, vehicles, etc..

Today video camera has been very popular, many mobile phones equipped with video cameras. We can use video camera to record a tennis ball fall from a height of a skyscraper. Video recording was used to determine how long the travel time to fall from skyscraper height to reach the ground. Some cameras record 15 frames per second (fps), some camera phones can do 30 frames per second, so we can get a very accurate calculation results.
The formula used is

H = ½ x gravity x time x time = h = ½. g. t2
Constant gravity = 9.80665 m/s2

The time is calculated when a tennis ball landed on the ground and deducted by the time when it was released at certain height. By using slow-motion menu we can see the detail of each frame that are recorded.

For example, the video record at 15 fps, the record shows the ball was released on 2nd frame at 3 seconds of record. Then the time recorded as the initial time is:

3 seconds + 2 /15 = 3 2 /15 sec

When tennis ball landed on the ground, video records at 6 second and on 10th frame, then the time that is recorded as the landing time is:

6 seconds + 10/15 = 6 10/15 seconds

So the travel time of a tennis ball when released from the skyscraper and landing to the ground is:

t = 6 10/15 - 3 2 /15 = 3 8 /15 seconds

Then the building height or distance covered by tennis ball as it fall from the skyscraper is:

S = ½ x 9.80665 x (3 8/15) 2 = 61.22 meters
The graph and table below gives an overview over the travel time in seconds of falling object, and it’s relationship to height.

 no t  (second) h (meter) 1 0.25 0.3 2 0.50 1.2 3 0.75 2.8 4 1.00 4.9 5 1.25 7.7 6 1.50 11.0 7 1.75 15.0 8 2.00 19.6 9 2.25 24.8 10 2.50 30.6 11 2.75 37.1 12 3.00 44.1 13 3.25 51.8 14 3.50 60.1 15 3.75 69.0 16 4.00 78.5 17 4.25 88.6 18 4.50 99.3 19 4.75 110.6 20 5.00 122.6