## Chapter 7

## Q. 7.12

## Q. 7.12

**FROM THE EARTH TO THE MOON**

**GOAL** Apply conservation of energy with the general form of Newton’s universal law of gravity.

**PROBLEM** In Jules Verne’s classic novel *From the Earth to the Moon*, a giant cannon dug into the Earth in Florida fired a spacecraft all the way to the Moon. (**a**) If the spacecraft leaves the cannon at escape speed, at what speed is it moving when 1.50 \times 10^5 \mathrm{~km} from the center of Earth? Neglect any friction effects. (**b**) Approximately what constant acceleration is needed to propel the spacecraft to escape speed through a cannon bore 1.00 \mathrm{~km} long?

**STRATEGY** For part (**a**), use conservation of energy and solve for the final speed v_f. Part (**b**) is an application of the timeindependent kinematic equation: solve for the acceleration a.

## Step-by-Step

## Verified Solution

(**a**) Find the speed at r=1.50 \times 10^5 \mathrm{~km}.

Apply conservation of energy:

\frac{1}{2} m v_i^2-\frac{G M_E m}{R_E}=\frac{1}{2} m v_f^2-\frac{G M_E m}{r_f}

Multiply by 2 \mathrm{/m} and rearrange, solving for v_f^2. Then substitute known values and take the square root.

\begin{aligned}v_f^2=& v_i^2+\frac{2 G M_E}{r_f}-\frac{2 G M_E}{R_E}=v_i^2+2 G M_E\left(\frac{1}{r_f}-\frac{1}{R_E}\right) \\v_f^2=&\left(1.12 \times 10^4 \mathrm{~m} / \mathrm{s}\right)^2+2\left(6.67 \times 10^{-11} \mathrm{~kg}^{-1} \mathrm{~m}^3 \mathrm{~s}^{-2}\right) \\& \times\left(5.98 \times 10^{24} \mathrm{~kg}\right)\left(\frac{1}{1.50 \times 10^8 \mathrm{~m}}-\frac{1}{6.38 \times 10^6 \mathrm{~m}}\right) \\v_f=& 2.39 \times 10^3 \mathrm{~m} / \mathrm{s}\end{aligned}

(**b**) Find the acceleration through the cannon bore, assuming it’s constant.

Use the time-independent kinematics equation:

\begin{aligned}v^2-v_0{ }^2 &=2 a \Delta x \\\left(1.12 \times 10^4 \mathrm{~m} / \mathrm{s}\right)^2-0 &=2 a\left(1.00 \times 10^3 \mathrm{~m}\right) \\a &=6.27 \times 10^4 \mathrm{~m} / \mathrm{s}^2\end{aligned}

**REMARKS** This result corresponds to an acceleration of over 6 000 times the free-fall acceleration on Earth. Such a huge acceleration is far beyond what the human body can tolerate.