Problem 80 - Physics

Question

A spectator, seated in the left field stands, is watching a baseball player \(1.9 \mathrm{~m}\) tall who is \(75 \mathrm{~m}\) away. On a TV screen, located \(3.0\mathrm{~m}\) from a person watching the game at home, the same player has a 0.12 -m image. Find the angular size of the player as seen by (a) the spectator watching the game live and (b) the TV viewer. (c) To whom does the player appear to be larger?

Step-by-Step Solution

Answer

The player appears larger to the TV viewer.

Step 1: Identify Angular Size for Live Spectator

Angular size can be calculated using the formula \( \theta = \frac{h}{d} \), where \( h \) is the height of the object and \( d \) is the distance from the observer. For the spectator:\[ \theta_{\text{live}} = \frac{1.9 \text{ m}}{75 \text{ m}} \]

Step 2: Calculate Angular Size for Live Spectator

Substitute the known values into the formula to calculate \( \theta_{\text{live}} \):\[ \theta_{\text{live}} = \frac{1.9}{75} = 0.0253 \text{ radians} \]

Step 3: Identify Angular Size for TV Viewer

Using the same formula, calculate angular size for the TV viewer who sees a 0.12 m image from 3.0 m away:\[ \theta_{\text{TV}} = \frac{0.12 \text{ m}}{3.0 \text{ m}} \]

Step 4: Calculate Angular Size for TV Viewer

Substitute the known values into the formula to calculate \( \theta_{\text{TV}} \):\[ \theta_{\text{TV}} = \frac{0.12}{3.0} = 0.04 \text{ radians} \]

Step 5: Compare Angular Sizes

Compare \( \theta_{\text{live}} \) and \( \theta_{\text{TV}} \): since \( \theta_{\text{TV}} = 0.04 \text{ radians} \) is greater than \( \theta_{\text{live}} = 0.0253 \text{ radians} \), the player appears larger to the TV viewer.

Key Concepts

Angular Size

Angular size is a concept that helps us understand how large an object appears to an observer, based on its physical size and distance from the observer. In simple terms, it is the angle that an object occupies in the viewer's field of vision.
This is an important aspect of perception because it doesn't matter how large the object is in real life; what matters is how big it looks to us. We can calculate angular size using the formula \( \theta = \frac{h}{d} \), where \( h \) represents the height of the object and \( d \) is the distance from the observer to the object.
For the live spectator, the baseball player who is 1.9 meters tall and 75 meters away has an angular size of about 0.0253 radians. Meanwhile, when viewed on a television screen, the image occupies a different size due to a shorter viewing distance, which can result in a different angular size. Understanding this difference is key in recognizing how perception differs based on context.

Optical Measurements

Optical measurements help us quantify how we perceive objects through the concept of angular size. By measuring angles, we can make predictions about how large objects appear, which can be particularly useful in various applications such as design, architecture, and television production.
When measuring an object's angular size, we rely on simple geometrical relationships. This involves understanding the object's size and placement relative to the observer's position.

• If you're watching a baseball game live, the player's size is measured from your seat to the field.
• On the other hand, watching the game on television changes the optical measurement due to the screen's size and your distance from it.

The role of optical measurements becomes clear in mathematical analysis and everyday experiences, giving us insights into how we visually interpret the world.

Image Perception

Image perception is fundamentally about how our brains interpret visual signals based on specific stimuli. Angular size plays a crucial role in this perception, as it influences how large or small an object appears in our view.
When watching a real-life event, like a baseball game, various factors such as lighting, angle, and distance affect our perception of images. However, when switching to a television, some of these factors change, leading to different perceptions. For instance:

• A player might seem smaller in person due to distance, yet larger on a screen when you sit closer.
• The brightness and color saturation on the television might also alter the perceived size.

Our perception is therefore affected by these properties, which alter our understanding and interpretation of an image's true form and size.

Viewing Distance

Viewing distance is a critical factor that influences how an object appears in terms of size and clarity. It's the physical space between you, the observer, and the object you're looking at.
In situations like watching a baseball game, where the distance between the viewer and the players can be quite large, the angular size tends to be smaller, making players look smaller than they actually are. Conversely, with a shorter distance as in watching through a TV, objects can appear larger and more detailed.
Here are some key points about viewing distance:

• Closer distances result in larger angular sizes, making objects appear larger.
• Greater distances reduce angular size, making objects seem smaller.
• Your viewing experience can be enhanced by adjusting your position relative to the object or screen.

Understanding how viewing distance impacts our experience can help you better appreciate the visual information you receive, whether in a stadium or on your couch.