## What is Bearing?

**Definition:** Bearing in physics is defined as the measurement of an angle in degrees, delineating the relationship between a reference direction, typically north, and the direction of an object. This angle measurement is of paramount importance in various fields, including navigation, cartography, and astronomy. Bearings serve as a critical bridge between abstract mathematical concepts and their practical application in the real world.

Bearing is the angular displacement or direction between cardinal points in an x – y plane. To find the direction of an object, we need to have a look at a plane or space. By looking at the four cardinal points from the perspective of a plane, it is obvious that we have x and y coordinates and we call them abscissa and ordinate respectively.

Additionally, we assign names to these four cardinal points or poles, and we call them north, south, west, and east. While calculating bearing, we use the symbol of θ to indicate the angular direction of an object. Here are some of the methods we apply to measure angular direction (θ):

- Applying compass
- Using a protractor on a map
- Simple calculation

## Solved Problems: How to Calculate Bearing in Physics

We have four cardinal points starting from North (N) to East (E), and from N to E covers an angle of 90^{0}. Depending on the direction of the object, the angle between two cardinal points is 90^{0}.

This makes movement from **North to East, and then to South 180 ^{0}**

From **North to East to South and then to West 270 ^{0}**.

And complete movement **from North, East, South, West, and back to North 360 ^{0}**

**After finding the direction of the object, this is how you write it:**

Assuming your θ is 60^{0}, **you now know that the direction θ=60 ^{0} is between North and East**

Thus, you write it as **N 60 ^{0}E**

Which means **sixty degrees from North direction towards East**

Now, here are few examples to guide you understand how to calculate bearing in physics

### Problem 1

A man walks 4 kilometres north and then walks 3km due East. What is his displacement from the starting point?

**Solution**

**Data**

We first extract our data from the above question

Let us assume that his initial position is the origin, O

He walks from his origin O to another point A due North = 4 kilometres

Additionally, He walks from his second position A to another location B due East = 3 kilometer

For us to find his displacement from the starting point, we calculate OB.

To calculate OB, we apply the formula below

OB^{2} = OA^{2} + AB^{2}

We have OB = ?, OA = 4km, and AB = 3km

Substituting the above values into the formula, we have

OB^{2} = 4^{2} + 3^{2} = 16 + 9 = 25

Therefore we take the square root of both sides, to find OB

OB = √25 = 5km

Now we can see that the displacement of the man from his initial position is 5km.

We need to also find the direction, θ

By applying the formula below, we have

tan θ = AB/OA = 3/4

Divide both sides by tan to find θ

θ = tan^{-1} (3/4) = tan^{-1} (0.75) = 36.9^{0}

We can now approximate θ as 37^{0} which is the direction of the man

θ = 37^{0}

Because 37^{0} is between North(N) and East (E), we can now say that the direction of the man is N 37^{0}E ( The man is thirty-seven degrees from North direction towards East)

Therefore, our final answer is as follows:

The displacement is **5km, N 37 ^{0}E**

### Problem 2

Find the location of A(3,-6) and B(7,9) with the reference to a point (0,0) and determine the bearing of A,

**Solution:**

**Data:**

Let us consider triangle OAB

When we trace our line on B axis from Origin(which is zero) to be on A(3,-6), we can see that X_{1} = 3 and this implies that OB = X_{1} = 3

Therefore, the modulus of OB= 3 units

modulus of AB= 6 units

We now apply the formula below

tan θ = OB/AB = 3/6

By making θ subject of the formula, we have

θ = tan^{-1} (0.5) = 26.6^{0}

Which can be approximated into

θ = 27^{0}

To find the bearing of A, we add 27^{0} to the 90^{0} which si being completed from North to East

Therefore, A = 90^{0} + 27^{0} = 117^{0}

Which is considered as E 117^{0}S

### Problem 3

A butterfly moves from a point X to a point Y, a distance of 15 meters in the direction 045^{0}. It then changes direction and moves to a point Z, in the direction 155^{0}. If the point Z is due east of X, find the distance of the point Z from the point Y.

**Solution**

**Data**

By applying the sine rule which is

X / Sinθ_{1} = Y / Sinθ_{2} = Z / Sinθ_{3}

and θ_{1} = 45^{0}, θ_{2} = 70^{0}, θ_{3} = 65^{0}, Z = 15m, and X = ?

we now substitute the above data into X / Sinθ_{1} = Z / Sinθ_{3}

X / Sin45^{0} = 15 / Sin65^{0}

We can now cross multiply the above equation

XSin65^{0} = 15 x Sin45^{0}

We now make X subject of the formula

X = (15 x Sin45^{0}) / (Sin65^{0}) = (15 x 0.71) / (0.91) = 10.65/0.91 = 11.7m

**Therefore the distance of point Z from point Y which is X = 11.7 meters**

## Explanation

**I. Types of Bearings**

** Brief Overview:** There are several types of bearings, each offering unique characteristics and applications. This section will focus on three primary types: true bearing, magnetic bearing, and relative bearing.

**II. True Bearing Calculation**

** Defining True Bearing:** True bearing signifies the angle between a fixed reference direction (usually north) and a specific target or object, measured in a clockwise direction. To calculate true bearing, one must consider the angular difference between the reference direction and the target’s direction.

** Calculation Formula:** The formula for calculating true bearing is as follows: True Bearing (TB) = Magnetic North (MN) ± Magnetic Variation (MV) ± Compass Reading (CR)

**I****I**I. Magnetic Bearing Calculation

**I**I. Magnetic Bearing Calculation

* Understanding Magnetic Bearing:* Magnetic bearing closely aligns with true bearing but necessitates adjustments based on the Earth’s magnetic field. This type is primarily employed in navigation, especially when working with compasses.

* Calculation Formula:* The formula for calculating magnetic bearing is as follows: Magnetic Bearing (MB) = True Bearing (TB) ± Magnetic Deviation (MD)

**IV. Relative Bearing Calculation**

*Exploring Relative Bearing* Relative bearing is predominantly used in navigation contexts where the point of reference is continuously changing, such as in tracking moving targets or vessels. It measures the angle between the forward direction of a moving object and a target’s direction.

*Calculation Formula* The formula for calculating relative bearing is as follows: Relative Bearing (RB) = True Bearing (TB) – Vessel’s Heading (VH)

**V. Practical Applications**

** Real-World Utility:** Understanding how to calculate bearings has direct relevance to numerous practical applications:

In marine and aviation navigation, precise bearing calculations are indispensable for determining the direction to a destination and avoiding collisions.*Navigation*:Land surveyors rely on bearing measurements to ascertain angles between survey points, a cornerstone for producing accurate maps and defining property boundaries.*Surveying*:Search and rescue operations utilize bearings to locate missing individuals or objects. By measuring the bearing to a known reference point, search teams can triangulate the position of the target.*Search and Rescue*:Astronomers employ bearings to locate celestial objects in the night sky. By calculating the bearing of stars or planets from a known reference point, astronomers can accurately identify celestial bodies.*Astronomy*:

**VI**. Tools and Technology

**VI**. Tools and Technology

** Instruments and Devices:** Advancements in technology and the development of specialized instruments have greatly enhanced the accuracy and efficiency of bearing calculations. Several tools and technologies play a significant role in this process:

A magnetic compass, a simple yet effective tool, is pivotal for measuring true and magnetic bearings.*Compass*:Gyrocompasses, renowned for their high accuracy, provide a reference to true north, eliminating the need for magnetic corrections. They are widely employed in marine and aviation navigation.*Gyrocompass*:GPS technology revolutionized navigation by providing precise position and bearing data using satellite signals.*GPS (Global Positioning System)*:Sextants, traditionally used in celestial navigation, measure the angle between a celestial body and the horizon to determine a vessel’s position.*Sextant*:Land surveyors utilize theodolites to measure horizontal and vertical angles accurately, enabling precise bearing calculations for various applications.*Theodolite*:

**VII. Common Challenges and Errors**

* Pitfalls to Avoid:* While bearing calculations are indispensable, they come with common challenges and potential errors. Recognizing and circumventing these pitfalls is crucial for ensuring accurate results.

Magnetic bearing calculations are highly sensitive to magnetic variation, which varies with the location on Earth. Failing to account for this variation can result in significant errors.*Magnetic Variation*:Magnetic compasses can be affected by magnetic deviation, caused by local magnetic fields on a vessel or aircraft. Calibrating the compass properly is vital to obtain accurate readings.*Compass Deviation*:Inaccurate measurements can stem from human error, parallax, or unstable conditions. Taking measurements with care and under stable conditions is essential to avoid errors.*Inaccurate Readings*:In the context of relative bearing, the point of reference may change frequently. It is crucial to keep track of these changes and adjust calculations accordingly.*Changing Reference Points*:

*You may like to read:*

How to Calculate Power in Physics

**Reference:**