Question

Supersonic jets are able to achieve maximum speeds of up to 3.0 × 10-6

c. Calculate
the percentage contraction in the length of a jet plane travelling at this speed.

Answer 1

[AI-generated answer — Bot GPT 3.5]

To calculate the percentage contraction in the length of the jet plane, we will use the formula for length contraction in special relativity:

\( L' = \frac{L}{\sqrt{1 - \frac{v^2}{c^2}}} \)

where:
L = original length of the jet plane
v = velocity of the jet plane (3.0 × 10-6 c)
c = speed of light

Plugging in the values:

\( L' = \frac{L}{\sqrt{1 - (3.0 \times 10^{-6})^2}} \)
\( L' = \frac{L}{\sqrt{1 - 9.0 \times 10^{-12}}} \)
\( L' = \frac{L}{\sqrt{1 - 9.0 \times 10^{-12}}} \)
\( L' = \frac{L}{\sqrt{0.999999999991}} \)
\( L' = \frac{L}{0.9999999999955} \)
\( L' \approx 0.9999999999955L \)

Now, to calculate the percentage contraction:

\( \text{Percentage contraction} = \frac{L - L'}{L} \times 100 \)
\( \text{Percentage contraction} = \frac{L - 0.9999999999955L}{L} \times 100 \)
\( \text{Percentage contraction} = \frac{0.0000000000045L}{L} \times 100 \)
\( \text{Percentage contraction} \approx 0.00000000045 \times 100 \)
\( \text{Percentage contraction} \approx 0.000000045 \% \)

Therefore, the jet plane will experience a contraction in its length of approximately 0.000000045%.