Question:
A rocket of length 18.0 m is moving at speed \(0.9c\) (where \(c\) is the speed of light) parallel to its own length, relative to the earth. The length of the rocket measured in meters by an observer on earth (rounded off to two decimal places) is
A rocket of length 18.0 m is moving at speed \(0.9c\) (where \(c\) is the speed of light) parallel to its own length, relative to the earth. The length of the rocket measured in meters by an observer on earth (rounded off to two decimal places) is
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Hint:
Use \(L = L_0\sqrt{1-v^2/c^2}\) with \(L_0=18.0\) m and \(v=0.9c\).
Use \(L = L_0\sqrt{1-v^2/c^2}\) with \(L_0=18.0\) m and \(v=0.9c\).
Updated On: Jul 28, 2026
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Correct Answer: 7.85
Solution and Explanation
Step 1: Understanding the Concept:
Length contraction says a moving object measured by an observer at rest looks shorter along its direction of motion than its length measured in its own rest frame. The rest frame length is called the proper length.
Step 2: Key Formula or Approach:
\[ L = L_0\sqrt{1 - \frac{v^2}{c^2}} \]
where \(L_0\) is the proper length (18.0 m, the length in the rocket's own frame) and \(v = 0.9c\) is its speed relative to the earth.
Step 3: Detailed Explanation:
\[ \frac{v^2}{c^2} = (0.9)^2 = 0.81 \]
\[ 1 - \frac{v^2}{c^2} = 1 - 0.81 = 0.19 \]
\[ \sqrt{0.19} = 0.4359 \]
\[ L = 18.0 \times 0.4359 = 7.846 \text{ m} \]
Final Answer:
Rounded to two decimal places, the observer on earth measures the rocket's length as 7.85 m. \[ \boxed{L = 7.85 \text{ m}} \]
Length contraction says a moving object measured by an observer at rest looks shorter along its direction of motion than its length measured in its own rest frame. The rest frame length is called the proper length.
Step 2: Key Formula or Approach:
\[ L = L_0\sqrt{1 - \frac{v^2}{c^2}} \]
where \(L_0\) is the proper length (18.0 m, the length in the rocket's own frame) and \(v = 0.9c\) is its speed relative to the earth.
Step 3: Detailed Explanation:
\[ \frac{v^2}{c^2} = (0.9)^2 = 0.81 \]
\[ 1 - \frac{v^2}{c^2} = 1 - 0.81 = 0.19 \]
\[ \sqrt{0.19} = 0.4359 \]
\[ L = 18.0 \times 0.4359 = 7.846 \text{ m} \]
Final Answer:
Rounded to two decimal places, the observer on earth measures the rocket's length as 7.85 m. \[ \boxed{L = 7.85 \text{ m}} \]
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