Problem 46 - University Physics with Modern Physics

Question

A photographer takes a photograph of a Boeing 747 airliner (length 70.7 \(\mathrm{m} )\) when it is flying directly overhead at an altitude of 9.50 \(\mathrm{km} .\) The lens has a focal length of 5.00 \(\mathrm{m} .\) How long is the image of the airliner on the film?

Step-by-Step Solution

Answer

The image of the airliner is 0.0372 m long.

Step 1: Understand the Problem

We need to calculate the size of the image of a Boeing 747 on film using the lens formula and the provided dimensions and distances.

Step 2: Convert Units

Convert the altitude of the airplane from kilometers to meters: 9.50 km = 9500 m.

Step 3: Use the Lens Formula

The lens magnification formula is given by: \( m = \frac{h_i}{h_o} = -\frac{f}{d_o - f} \), where \( h_i \) is the image height, \( h_o \) is the object height (70.7 m), \( f \) is the focal length (5.00 m), and \( d_o \) is the object distance (9500 m).

Step 4: Calculate the Object Distance Beyond the Focal Length

Substitute the values into the magnification formula: \( h_i = h_o \cdot \frac{-f}{d_o - f} = 70.7 \cdot \frac{-5}{9500 - 5} \).

Step 5: Solve for Image Height

Calculate \( h_i \):\[ h_i = 70.7 \cdot \frac{-5}{9495} \approx -0.0372 \ m \].The negative sign indicates the image is inverted, but the length is 0.0372 m.

Key Concepts

Lens Formula

The lens formula is an essential tool in optics, crucial for understanding how lenses form images. It relates three important variables:

• The object distance (\( d_o \)), which is how far the object is from the lens.
• The image distance (\( d_i \)), which is the distance from the lens to the projected image.
• The focal length (\( f \)) of the lens, a fixed property depending on the lens's curvature and the refractive index.

These values are linked by the equation:\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}.\]This formula is fundamental for finding out how lenses adjust distances to produce clear images. It can be rearranged to solve for any of the three variables if the other two are known. In this scenario, knowing the focal length and object distance helps us figure out what the resulting image distance will be.

Magnification

Magnification in optics allows us to quantify how much larger or smaller the image appears compared to the object itself. The magnification (\( m \)) is calculated using the ratio:\[ m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}, \]where:

• \( h_i \) is the image height.
• \( h_o \) is the object height.
• \( d_i \) and \( d_o \) are the image and object distances, respectively.

In our exercise, this formula helps determine how the Boeing 747's image size compares to its actual size. The negative sign indicates that the image is inverted. For this practical application, magnification shows how the large Boeing 747 can appear as a much smaller image on film.

Image Formation

Image formation via lenses is all about how light bends and converges to produce a replica of the object. In our example, the image of the Boeing 747 is formed on the camera's film. This involved calculating precisely where the rays converge to pinpoint the image location, whether on the same side or the opposite side of the lens. Key aspects to keep in mind include:

• How light travels through the lens.
• The focus point where rays converge.
• Whether the image is real (can be projected) or virtual (cannot be).
• Image orientation, whether upright (positive height) or inverted (negative height).

For the Boeing 747, using the calculated values helps us understand the scale and orientation of the image captured on film, with inversion denoting a real image.

Focal Length

Focal length is a critical attribute of lenses; it dictates how strongly a lens can bend light. It's defined as the distance between the lens and its focal point, where parallel rays of light meet after passing through the lens. A shorter focal length means light converges more quickly, producing a larger, closer image. Conversely, a longer focal length results in a smaller, more distant image. Here are some important things about focal length:

• It affects the lens's field of view.
• Affects depth of field, influencing how much of an image appears in focus.

For our problem, the lens used has a focal length of 5.00 m. This value guides how we use both the lens formula and magnification factors to find the Boeing 747's image size on film. Understanding focal length is essential for photographers to assess and manipulate the composition and clarity of their images.