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Chapter 35: Problem 22

Find the value of \(g\), the gravitational acceleration at Earth's surface, in light-years per year, to three significant figures.

Short Answer

Expert verified
Answer: 1.13 x 10^-16 ly/yr²

Step by step solution

01

Determining Known Values and Conversion Factors

The known values required are: - \(g\) (Earth's gravitational acceleration) in meters per second squared (m/s^2): \(g = 9.81\,\text{m/s}^2\). - \(c\) (speed of light) in meters per second (m/s): \(c = 3.00\times10^8\,\text{m/s}\). - \(t\) (number of seconds in a year): \(t = 3.15\times10^7\,\text{s}\). With these known values, we can come up with the necessary conversion factors: - Meters to light-years: \(1\,\text{light-year} = \frac{c \times t}{1\,\text{year}}\). - Seconds to years: \(1\,\text{year} = 3.15 \times 10^7\,\text{s}\). Step 2:
02

Converting Earth's Gravitational Acceleration to Light-Years Per Year

We will first find the conversion factor for meters to light-years: \(1\,\text{light-year} = \frac{c \times t}{1\,\text{year}} = \frac{3.00\times10^8\,\text{m/s} \times 3.15\times10^7\,\text{s}}{1\,\text{year}} = 9.46\times10^{15}\,\text{m}\). Now, we can convert the value of \(g\) from meters per second squared to light-years per year squared: \(g_{\text{ly/yr}^2} = \left(\frac{9.81\,\text{m/s}^2}{9.46\times10^{15}\,\text{m/ly}}\right)\times\left(\frac{1\,\text{year}}{3.15\times10^7\,\text{s}}\right)^2\) Step 3:
03

Rounding the Result to Three Significant Figures

Evaluating the previous expression, we get: \(g_{\text{ly/yr}^2} \approx 1.1251 \times 10^{-16}\,\text{ly/yr}^2\) To round the result of \(g\) to three significant figures, we get: \(g_{\text{ly/yr}^2} \approx 1.13 \times 10^{-16}\,\text{ly/yr}^2\) So, the value of Earth's gravitational acceleration in light-years per year squared, to three significant figures, is \(1.13 \times 10^{-16}\,\text{ly/yr}^2\).

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A spaceship is traveling at two-thirds of the speed of light directly toward a stationary asteroid. If the spaceship turns on it headlights, what will be the speed of the light traveling from the spaceship to the asteroid as observed by a) someone on the spaceship? b) someone on the asteroid?

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