A certain automobile is 6 m long if rest. If it is measured to be `4//5` as long, its speed is

Options

A. 0.1c
B. 0.3c
C. 0.6c
D. 0.8c

Answer

To solve the problem step by step, we need to apply the concept of length contraction from the theory of relativity.

### Step 1: Understand the problem
The automobile has a proper length \( L_0 = 6 \) m when at rest. When it is moving, its length is measured to be \( \frac{4}{5} \) of its proper length.

### Step 2: Write the formula for length contraction
According to the theory of relativity, the length \( L \) of an object moving at speed \( v \) is given by:
\[ L = \frac{L_0}{\gamma} \]
where \( \gamma \) (the Lorentz factor) is defined as:
\[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]
Here, \( c \) is the speed of light.

### Step 3: Set up the equation
Given that the moving length \( L \) is \( \frac{4}{5} L_0 \), we can write:
\[ \frac{4}{5} L_0 = \frac{L_0}{\gamma} \]

### Step 4: Cancel \( L_0 \)
Since \( L_0 \) is not zero, we can cancel it from both sides:
\[ \frac{4}{5} = \frac{1}{\gamma} \]

### Step 5: Express \( \gamma \)
Taking the reciprocal gives:
\[ \gamma = \frac{5}{4} \]

### Step 6: Substitute \( \gamma \) into the Lorentz factor equation
Now, substituting \( \gamma \) into the equation for \( \gamma \):
\[ \frac{5}{4} = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]

### Step 7: Square both sides
Squaring both sides gives:
\[ \left(\frac{5}{4}\right)^2 = \frac{1}{1 - \frac{v^2}{c^2}} \]
\[ \frac{25}{16} = \frac{1}{1 - \frac{v^2}{c^2}} \]

### Step 8: Cross-multiply to solve for \( v^2 \)
Cross-multiplying gives:
\[ 25(1 - \frac{v^2}{c^2}) = 16 \]
Expanding this:
\[ 25 - 25\frac{v^2}{c^2} = 16 \]

### Step 9: Rearranging the equation
Rearranging gives:
\[ 25 - 16 = 25\frac{v^2}{c^2} \]
\[ 9 = 25\frac{v^2}{c^2} \]

### Step 10: Solve for \( v^2 \)
Now, solving for \( v^2 \):
\[ \frac{v^2}{c^2} = \frac{9}{25} \]
\[ v^2 = \frac{9}{25}c^2 \]

### Step 11: Take the square root to find \( v \)
Taking the square root gives:
\[ v = \frac{3}{5}c \]

### Step 12: Substitute \( c \) for numerical value
Using \( c \approx 3 \times 10^8 \) m/s:
\[ v = \frac{3}{5} \times 3 \times 10^8 \text{ m/s} = 1.8 \times 10^8 \text{ m/s} \]

### Final Answer
The speed of the automobile is \( 1.8 \times 10^8 \) m/s.

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