Error in Measurement A degree may seem like a very small...

Error in Measurement A degree may seem like a very small unit, but an error of one degree in measuring an angle may be very significant. For example, suppose a laser beam directed toward the visible center of the moon misses its assigned target by $30.0 "$. How far is it (in miles) from its assigned target? Take the distance from the surface of Earth to that of the moon to be $234,000$ mi. (Source: A Sourcebook of Applications of School Mathematics by Donald Bushaw et al.)

Key Concepts

Angular Measurement

Angular measurement involves quantifying angles, which are typically expressed in degrees, minutes, or seconds. This concept is fundamental in many fields such as astronomy, physics, and engineering, where even very small angle differences can correspond to large linear distances when projected over vast spaces.

Unit Conversion for Angles

Converting between different units of angular measure is essential when working with precise measurements. Conversions, such as from degrees or arc seconds to radians, are necessary because many mathematical formulas, especially those in trigonometry and calculus, require the use of radians to relate angles to arc lengths or other quantities.

Small Angle Approximation

The small angle approximation is used when angles are sufficiently small that their sine, tangent, and the angle itself (in radians) can be considered nearly equal. This simplification, typically applied in physics and astronomy, allows for easier estimation of linear displacements from angular errors over large distances.

Error Propagation in Measurements

Error propagation examines how uncertainties in measured quantities, such as an angle, affect the accuracy of calculated results. In contexts where a small error in an angle measurement is magnified over a large distance, understanding and calculating this propagation is vital for assessing the reliability of the measurement and subsequent predictions.

Transcript

00:01 This problem explores using pythagorean theorem and exploring how a very small change in a data can result in a very large error, right, in over a very large distance.

00:20 So i'm using a made -up example here.

00:22 Let's say you're on this strange planet over here, and you want to shoot an entire laser beam to blow up this small blue planet over here on the side.

00:30 Now the distance from the center to center or wherever you're shooting your beam from and this small circle over here, which is our second planet, is going to be 100 ,000 miles.

00:45 Let's say.

00:47 And our data is going to be what's 500 seconds.

00:51 It's a bit of strange measure of degrees.

00:55 So the way the degrees are split is actually degrees and then plus a what's known as minutes.

01:02 And then plus what's known as seconds.

01:06 So this would be like a degrees and b minutes and c seconds.

01:13 And the way it works is the minutes are divided by 60.

01:17 So this is a factor of 60 here, or 1 over 60 rather.

01:22 And the seconds are divided by 1 over 3 ,600, right? because it's just 60 times 60.

01:29 This becomes 3 ,600.

01:31 So it's a bit strange the way that the question is worded, but it's 500 seconds is going to be 500 over 36, and that's actually our full degree measure.

01:42 So let me go ahead and plug that into a calculator, and let's see what we get there.

01:51 So our theta comes out to in degrees, right, because this is in seconds.

01:57 So we want it in degrees.

01:59 So we convert this to 0 .139.

02:07 I'm going to round a little bit here.

02:08 So that is our value in degrees.

02:12 So now that we know data, we can find what our triangle looks like, right? so let's draw out our triangle.

02:21 So let's say here is your target x marks the spot...