Question 7.12: FROM THE EARTH TO THE MOON GOAL Apply conservation of energy...
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.
(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.