Question

Octavio, traveling at a speed of \(0.60 c,\) passes Tracy and her barn. Tracy, who is at rest with respect to her barn, says that the barn is 16 m long in the direction in which Octavio is traveling, \(4.5 \mathrm{m}\) high, and $12 \mathrm{m}$ deep. (a) What does Tracy say is the volume of her barn? (b) What volume does Octavio measure?

Short answer

Answer: The volume of the barn measured by Tracy is 864 m³, while Octavio measures the volume as 691.2 m³.

Step-by-step solution

Calculate the volume of the barn according to Tracy

Since Tracy is at rest relative to the barn, we can use the given dimensions to calculate the barn's volume: Volume = Length × Width × Height Tracy's measured volume = \(16\text{m} \times 4.5\text{m} \times 12\text{m}\)

Calculate Tracy's measured volume

Plugging the values into the equation, we get the volume as measured by Tracy: Tracy's measured volume = \(16\text{m} \times 4.5\text{m} \times 12\text{m} = 864\text{m}^3\)

Apply length contraction to the barn's length

To find the length of the barn as measured by Octavio, we need to use the length contraction equation: Contracted length = Length / \(\gamma\) Here, \(\gamma = \frac{1}{\sqrt{1 - (v/c)^2}}\), and Octavio's velocity (v) is given as \(0.6c\).

Calculate the gamma factor

We can calculate \(\gamma\) using the given velocity: \(\gamma = \frac{1}{\sqrt{1 - (0.6c/c)^2}} = \frac{1}{\sqrt{1 - 0.36}}\)

Compute the gamma factor

Solve the expression for the gamma factor: \(\gamma = \frac{1}{\sqrt{0.64}} = \frac{1}{0.8} = 1.25\)

Calculate the contracted length

Now, we can find the contracted length using the length contraction equation: Contracted length = \(\frac{16\text{m}}{1.25} = 12.8\text{m}\)

Calculate the volume of the barn according to Octavio

Lastly, we can calculate the volume of the barn as measured by Octavio by multiplying the contracted length with the unchanged height and depth: Octavio's measured volume = \(12.8\text{m} \times 4.5\text{m} \times 12\text{m}\)

Find Octavio's measured volume

Solve for the volume as measured by Octavio: Octavio's measured volume = \(12.8\text{m} \times 4.5\text{m} \times 12\text{m} = 691.2\text{m}^3\) To summarize the results: (a) Tracy says the volume of her barn is \(864\text{m}^3\) (b) Octavio measures the volume of the barn as \(691.2\text{m}^3\)