Ch 34: Ray Optics

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 34: Ray Optics

Problem 73

Chapter 34, Problem 73

CALC A wildlife photographer with a 200-mm-focal-length telephoto lens on his camera is taking a picture of a rhinoceros that is 100 m away. Suddenly, the rhino starts charging straight toward the photographer at a speed of 5.0 m/s. What is the speed, in μm/s, image of the of the rhinoceros? Is the image moving toward or away from the lens?

Verified step by step guidance

Step 1: Start by identifying the relevant formula for the magnification of a lens. The magnification (M) is given by the ratio of the image distance (v) to the object distance (u): M = v / u. The rate of change of the image distance can be related to the rate of change of the object distance using the lens equation.

Step 2: Recall the lens equation: (1 / f) = (1 / v) - (1 / u), where f is the focal length of the lens, v is the image distance, and u is the object distance. Differentiate this equation with respect to time (t) to relate the rates of change of v and u.

Step 3: Differentiating the lens equation with respect to time gives: 0 = (-1 / u²) * (du/dt) + (1 / v²) * (dv/dt). Here, du/dt is the rate of change of the object distance (the speed of the rhino, which is -5.0 m/s since the rhino is moving toward the lens), and dv/dt is the rate of change of the image distance (what we need to find).

Step 4: Solve for dv/dt in terms of du/dt, u, and v. Rearrange the differentiated equation to get: dv/dt = (v² / u²) * (du/dt). Substitute the known values: u = 100 m, f = 200 mm (convert to meters: 0.2 m), and du/dt = -5.0 m/s. Use the lens equation to first calculate v.

Step 5: Once v is determined using the lens equation, substitute v, u, and du/dt into the expression for dv/dt. Convert the result for dv/dt from m/s to μm/s (1 m/s = 1,000,000 μm/s). Finally, determine whether the image is moving toward or away from the lens based on the sign of dv/dt.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Focal Length and Magnification

Focal length is the distance from the lens to the image sensor when the subject is in focus. A longer focal length, like 200 mm, allows for greater magnification of distant subjects. Understanding how focal length affects image size and clarity is crucial for analyzing how the image of the rhinoceros will appear as it approaches the camera.

Relative Motion

Relative motion refers to the change in position of an object as observed from a particular frame of reference. In this scenario, the rhinoceros is moving toward the photographer, which affects how quickly the image of the rhino moves across the camera's sensor. This concept is essential for calculating the speed of the image as perceived by the photographer.

Image Speed Calculation

The speed of the image can be calculated using the formula that relates the object's speed to the image speed through the lens. Specifically, the image speed is influenced by the ratio of the object distance to the focal length. This calculation is necessary to determine how fast the image of the rhinoceros is moving in relation to the lens as it charges toward the photographer.

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