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Chapter 16: Problem 47

Image of an object approaching a convex mirror of radius of curvature \(20 \mathrm{~m}\) along its optical axis is observed to move from \(\frac{25}{3} \mathrm{~m}\) to \(\frac{50}{7} \mathrm{~m}\) in 30 seconds. What is the speed of the object in \(\mathrm{km}\) per hour ?

Short Answer

Expert verified
The speed of the object is approximately 1.2 km/h.

Step by step solution

01

Understanding the mirror equation

The mirror equation for a convex mirror is given by \(\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\), where \(f\) is the focal length, \(v\) is the image distance (with a negative value because images formed by convex mirrors are virtual), and \(u\) is the object distance (which should be taken as positive in this context). Because the radius of curvature \(R\) is 20 m, the focal length \(f\) is half that, at \(f = \frac{R}{2} = 10 \mathrm{~m}\).
02

Convert the image distances to object distances

Using the mirror equation, calculate the object distances corresponding to the image distances \(\frac{25}{3}\) m and \(\frac{50}{7}\) m. Image distances are negative for convex mirrors, so plug in \(v = -\frac{25}{3}\) m and \(v = -\frac{50}{7}\) m separately into the mirror equation to find the corresponding \(u\) values.
03

Calculate the change in object distance

Determine the change in object distance by taking the difference between the two object distances obtained from Step 2.
04

Calculate the speed of the object in meters per second

Use the formula for speed \(speed = \frac{\text{distance}}{\text{time}}\) to calculate the object's speed in meters per second, using the change in object distance from Step 3 and the given time of 30 seconds.
05

Convert the speed to kilometers per hour

Convert the speed from meters per second to kilometers per hour by using the unit conversion factor \(1 \text{ m/s} = 3.6 \text{ km/h}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mirror Equation
The mirror equation is a fundamental concept in optics that relates the distances of object and image to the focal length of the mirror. Specifically for a convex mirror, which forms virtual images, the equation is given as \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \), where \( f \) is the focal length, \( v \) is the image distance, and \( u \) is the object distance. Importantly, in this context the image distance \( v \) for convex mirrors is taken as negative since convex mirrors always form images that are virtual and upright. The focal length \( f \) is positive for convex mirrors and is half the radius of curvature according to \( f = \frac{R}{2} \). When you know the object's distance and the mirror's focal length, you can calculate the image distance using this equation, and vice versa.

This equation is vital because it helps us understand how images are formed in convex mirrors and under what conditions. Recognizing the convention that the physical quantities like \( f \) and \( v \) might adopt positive or negative values depending on the scenario is key to correctly applying the mirror equation.
Object Distance
In optics, the object distance, denoted as \( u \) in the mirror equation, represents the distance from the mirror to the object being viewed. For mirrors, object distance is typically considered positive, reflecting the fact that objects are placed in front of the mirror. When we analyze problems involving convex mirrors, understanding the object distance is crucial because it influences where the image will be formed and how large it will appear relative to the object.

Computing the object distance through the mirror equation requires an understanding of both the image distance and focal length. If either the image distance \( v \) or the focal length \( f \) changes, it affects the position of the object to maintain the relationship defined by the mirror equation. This is particularly helpful in exercises where the object's movement is being studied, as in the given problem where we calculate the object distance at two different points in time to determine the change and thereby the speed of the object.
Image Distance
The image distance, represented by \( v \) in the mirror equation, refers to the distance along the optical axis from the mirror to the location where an image is formed. For convex mirrors, this distance is always negative because the image is virtual (meaning it cannot be projected onto a screen) and appears behind the mirror. Knowing the image distance is a powerful tool in predicting how an object's position relative to the mirror will affect the appearance of its image.

While working through optics problems, it's essential to remember that image distance not only depends on the object's location but also on the characteristics of the mirror, such as its curvature and focal length. In the context of our exercise, understanding the relationship between object and image distances allows us to trace the path of the object's image in the mirror as it moves.
Speed Calculation
Speed calculation is a fundamental aspect of physics involving motion. It determines how fast an object is moving over a certain distance. The basic formula to calculate speed is \(speed = \frac{\text{distance}}{\text{time}}\), which requires knowledge of the distance traveled and the time taken to traverse that distance. This formula can be applied to a myriad of scenarios, including objects reflected in mirrors.

In the example problem, we used the change in object distance obtained by understanding the mirror equation to calculate the speed of the object as it approaches the convex mirror. We then converted the speed from meters per second (m/s) to kilometers per hour (km/h) to provide a more conventional unit of speed. It's crucial to pay strict attention to unit conversions and to understand that in the context of mirror reflections, the 'distance' traveled can refer to the changing distance between the object and its image.

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