A fire truck 10.0 meters long needs to fit into a garage...

Name: Problem 29, Chapter 35: University Physics with Modern Physics
Uploaded: 2021-11-03T15:49:59-08:00
Duration: 1 min 1 s
Channel: Narayan Hari
Description: A fire truck 10.0 meters long needs to fit into a garage 8.00 meters long (at least temporarily). How fast must the fire truck be going to fit entirely inside the garage, at least temporarily? How long does it take for the truck to get inside the garage, from a) the garage's point of view? b) the fire truck's point of view?

A fire truck 10.0 meters long needs to fit into a garage 8.00 meters long (at least temporarily). How fast must the fire truck be going to fit entirely inside the garage, at least temporarily? How long does it take for the truck to get inside the garage, from a) the garage's point of view? b) the fire truck's point of view?

A fire truck 10.0 meters long needs to fit into a garage 8.00 meters long (at least temporarily). How fast must the fire truck be going to fit entirely inside the garage, at least temporarily? How long does it take for the truck to get inside the garage, from
a) the garage's point of view?
b) the fire truck's point of view?

Key Concepts

Special Relativity

Special relativity is the fundamental framework that describes how measurements of time, length, and other physical quantities differ between observers in different inertial frames, particularly when objects move at speeds comparable to the speed of light. This theory underpins the phenomena of length contraction and time dilation, which are critical for understanding how a moving fire truck might appear shortened to a stationary observer.

Length Contraction

Length contraction is a consequence of special relativity which states that an object moving relative to an observer will appear shorter along the direction of motion. In this problem, the fire truck's proper length (its length in its own rest frame) is 10.0 meters, but due to its high speed relative to the garage, its measured length in the garage frame can be contracted to 8.00 meters, allowing it to fit entirely inside the garage.

Proper Length

The proper length is the length of an object measured in the object’s own rest frame, where it is not moving relative to the observer. It is considered the maximum or true length of the object. Here, the fire truck’s proper length is 10.0 meters, and this value is used as the baseline when applying the length contraction formula to determine the effective length observed in another frame.

Reference Frames

A reference frame refers to the coordinate system or viewpoint from which an observer makes measurements of physical quantities such as time and distance. In this problem, the garage’s frame and the fire truck’s frame are two different reference frames. Due to their relative motion, events such as the fire truck entering the garage are timed differently in each frame, illustrating how simultaneity and duration are not absolute but depend on the observer's state of motion.

Lorentz Factor

The Lorentz factor, denoted by ? (gamma), is a crucial component in the equations of special relativity that quantifies the effects of time dilation and length contraction. It is defined by the equation ? = 1/?(1 - v²/c²), where v is the relative velocity between observers and c is the speed of light. In this problem, the Lorentz factor is used to calculate how fast the fire truck must travel so that its contracted length in the garage’s frame meets the necessary dimension constraints, as well as to relate time intervals in the two different frames.

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