Question

For distant objects, the angular size in degrees can be approximated as $57^{\circ} \times w / d$, where $w$ is the width of the object and $d$ is its distance. What is the angular separation of the headlights on a car $10 \mathrm{km}$ away if the headlights are $1.2 \mathrm{m}$ apart?

Name: Problem 73, Chapter 19: Physics: A Conceptual World View
Uploaded: 2020-06-24T13:46:31-08:00
Duration: 1 min 3 s
Channel: Sanjeev Kumar
Description: For distant objects, the angular size in degrees can be approximated as 57^∘ ×w / d, where w is the width of the object and d is its distance. What is the angular separation of the headlights on a car 10 km away if the headlights are 1.2 m apart?

   For distant objects, the angular size in degrees can be approximated as $57^{\circ} \times w / d$, where $w$ is the width of the object and $d$ is its distance. What is the angular separation of the headlights on a car $10 \mathrm{km}$ away if the headlights are $1.2 \mathrm{m}$ apart?

Physics: A Conceptual World View

Larry D.… 7th Edition

Chapter 19, Problem 73

Instant Answer

Solved by Expert Sanjeev Kumar

06/24/2020

Step 1

Here, the distance is given in kilometers and the width is given in meters. So, we convert the distance from kilometers to meters. We know that 1 kilometer is equal to 1000 meters. So, 10 kilometers is equal to $10 \times 1000 = 10000$ meters. Show more…

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For distant objects, the angular size in degrees can be approximated as $57^{\circ} \times w / d$, where $w$ is the width of the object and $d$ is its distance. What is the angular separation of the headlights on a car $10 \mathrm{km}$ away if the headlights are $1.2 \mathrm{m}$ apart?

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Key Concepts

Angular Measurement

Angular measurement involves quantifying the size of an angle, typically in units such as degrees or radians. In the context of distant objects, angular size tells us how large or separated objects appear from a given viewpoint, even though their actual sizes and separations may be much larger.

Unit Consistency

Ensuring unit consistency is crucial in calculations involving physical dimensions. When applying formulas such as the angular size approximation, all measurements must be in compatible units so that the computed angle is accurate and meaningful.

Small-Angle Approximation

This concept is used when the angular size or separation of an object is very small, and it allows one to approximate the angle (in radians or degrees) as the ratio of the object's linear size to its distance. The approximation simplifies trigonometric functions like sine or tangent to the angle itself when measured in radians, and scaled appropriately when using degrees.

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