How to Calculate Bearing in Physics

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 90⁰. Depending on the direction of the object, the angle between two cardinal points is 90⁰.

This makes movement from North to East, and then to South 180⁰

From North to East to South and then to West 270⁰.

And complete movement from North, East, South, West, and back to North 360⁰

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

Assuming your θ is 60⁰, you now know that the direction θ=60⁰ is between North and East

Thus, you write it as N 60⁰E

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

OB² = OA² + AB²

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

Substituting the above values into the formula, we have

OB² = 4² + 3² = 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) = 36.9⁰

We can now approximate θ as 37⁰ which is the direction of the man

θ = 37⁰

Because 37⁰ is between North(N) and East (E), we can now say that the direction of the man is N 37⁰E ( The man is thirty-seven degrees from North direction towards East)

Therefore, our final answer is as follows:

The displacement is 5km, N 37⁰E

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 X₁ = 3 and this implies that OB = X₁ = 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) = 26.6⁰

Which can be approximated into

θ = 27⁰

To find the bearing of A, we add 27⁰ to the 90⁰ which si being completed from North to East

Therefore, A = 90⁰ + 27⁰ = 117⁰

Which is considered as E 117⁰S

Problem 3

A butterfly moves from a point X to a point Y, a distance of 15 meters in the direction 045⁰. It then changes direction and moves to a point Z, in the direction 155⁰. 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 / Sinθ₁ = Y / Sinθ₂ = Z / Sinθ₃

and θ₁ = 45⁰, θ₂ = 70⁰, θ₃ = 65⁰, Z = 15m, and X = ?

we now substitute the above data into X / Sinθ₁ = Z / Sinθ₃

X / Sin45⁰ = 15 / Sin65⁰

We can now cross multiply the above equation

XSin65⁰ = 15 x Sin45⁰

We now make X subject of the formula

X = (15 x Sin45⁰) / (Sin65⁰) = (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

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

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)

III. Magnetic Bearing Calculation

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

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

How it affects us in the Real-World: Understanding how to calculate bearings has direct relevance to numerous practical applications:

Navigation: In marine and aviation navigation, precise bearing calculations are indispensable for determining the direction to a destination and avoiding collisions.
Surveying: Land surveyors rely on bearing measurements to ascertain angles between survey points. This is a cornerstone for producing accurate maps and defining property boundaries.
Search and Rescue: 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.
Astronomy: 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.

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:

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

VII. Common Challenges and Errors

While bearing calculations are indispensable, they come with common challenges and potential errors. Recognizing and circumventing these pitfalls is important for ensuring accurate results.

Magnetic Variation: 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.
Compass Deviation: 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. No
Inaccurate Readings: Inaccurate measurements can stem from human error, parallax, or unstable conditions. Taking measurements with care and under stable conditions is essential to avoid errors.
Changing Reference Points: 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.

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Reference:

Wikipedia