Chapter 8: Problem 8

The value of escape velocity on the surface of the earth is (1) $11.27 \mathrm{~m} \mathrm{~s}^{-1}$ (2) $11.27$ mile $\mathrm{s}^{-1}$ (3) $7 \mathrm{~m} \mathrm{~s}^{-1}$ (4) 7 mile $s^{-1}$

Short Answer

Expert verified

Answer: (1) 11.27 km/s

Step by step solution

Write down the escape velocity formula

The formula for escape velocity is given by: $$v_e = \sqrt{\frac{2GM}{R}}$$, where $v_e$ is the escape velocity, $G$ is the gravitational constant, $M$ is the mass of the body (Earth), and $R$ is its radius.

Input the known values

We need to substitute the values of the Earth's mass and radius, along with the gravitational constant, into the formula. The mass of Earth, $M = 5.97 \times 10^{24} \mathrm{~kg}$ The radius of Earth, $R = 6.37 \times 10^6 \mathrm{~m}$ Gravitational constant, $G = 6.674 \times 10^{-11} \mathrm{\frac{Nm^2}{kg^2}}$

Calculate escape velocity

Substitute these values into the escape velocity formula: $$v_e = \sqrt{\frac{2 \times 6.674 \times 10^{-11} \times 5.97 \times 10^{24}}{6.37 \times 10^6}}$$

Solve the equation

Solving the equation, we get: $$v_e = \sqrt{\frac{7.97 \times 10^{14}}{6.37 \times 10^6}} = \sqrt{1.251 \times 10^8}$$ $$v_e \approx 11,186 \mathrm{~m} \mathrm{~s}^{-1}$$

Convert to the suitable option

We notice that this value is close to $11.27 \mathrm{~m} \mathrm{~s}^{-1}$, and none of the other options match this value. So, the correct answer is: (1) $11.27 \mathrm{~m} \mathrm{~s}^{-1}$

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The value of escape velocity on the surface of the earth is (1) \(11.27 \mathrm{~m} \mathrm{~s}^{-1}\) (2) \(11.27\) mile \(\mathrm{s}^{-1}\) (3) \(7 \mathrm{~m} \mathrm{~s}^{-1}\) (4) 7 mile \(s^{-1}\)

Short Answer

Step by step solution

Write down the escape velocity formula

Input the known values

Calculate escape velocity

Solve the equation

Convert to the suitable option

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