Question 7.13: Goal Apply conservation of energy with the general form of N...
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 × 10⁵ km from the center of Earth? Neglect any friction effects. (b) Approximately what constant acceleration is needed to propel the spacecraft to this speed through a cannon bore a kilometer long?
Strategy For part (a), use conservation of energy and solve for the final speed v_f . Part (b) is an application of the time-independent kinematic equation: solve for the acceleration a.
(a) Find the speed at r = 1.50 × 10⁵ km.
Apply conservation of energy:
\frac{1}{2}{mv_i}^2 – \frac{GM_Em}{R_E} = \frac{1}{2}{mv_f}^2 – \frac{GM_Em}{r_f}
Multiply by 2/m and rearrange, solving for {v_f}^2. Then substitute known values and take the square root.
{v_f}^2 = {v_i}^2 + \frac{2GM_E}{r_f} – \frac{2GM_E}{R_E} = {v_i}^2 + 2GM_E (\frac{1}{r_f} – \frac{1}{R_E})
{v_f}^2 = (1.12 × 10^4 m/s)^2 + 2(6.67 × 10^{-11} kg^{-1} m^3s^{-2})
× (5.98 × 10^{24} kg)(\frac{1}{1.50 × 10^8 m} – \frac{1}{6.37 × 10^6 m})
v_f = 2.35 × 10^3 m/s
(b) Find the acceleration through the cannon bore, assuming that it’s constant.
Use the time-independent kinematics equation:
v^2 – {v_0}^2 = 2aΔx
(1.12 × 10^4 m/s)^2 – 0 = 2a(1.00 × 10^3 m)
a = 6.27 × 10^4 m/s^2
Remark 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.