Universal Gravity

The James Cameron film Avatar (2009) takes place on a moon called Pandora around another planet.  In the film people are able to able to transport their minds into artificially constructed bodies, avatars, that look like the moon’s native people.  In this scene, our hero Jake Sully, is in his avatar but gets separated from his group and has to deal with Pandora’s native wildlife.

What distance does Jake Sully fall before hitting the water?

Solution: If this was the planet Earth, then this would just be a problem in projectile motion. However, we know that this doesn’t take place on Earth, so we need to use Newton’s Law of Universal Gravity to determine the local gravitational acceleration on Pandora. From the internet we can find that one estimate for the mass and size of Pandora is 45% of Earth’s mass and 75% of Earth’s size. That would give it a mass of MP = 0.45 (5.98×1024 kg) =  2.7×1024 kg and a radius of rP = 0.75(6.37×106 m) = 4.8×106 m. So from Newton’s Law of Universal Gravity we would have

F = G{Mm\over{r^2}} \implies g_P = {F/{m}} = G{M_P\over{r_P^2}}

g_P = 6.67 \times 10^{-11} {Nm\over{kg^2}} \left( {4.3 \times 10^{24} kg \over{(5.72 \times 10^6 m)^2}}\right) = 7.8 m/s^2

Now that we have the local gravitational acceleration on Pandora we can solve the problem like any projectile motion problem. We can time the fall from the clip and see it is about 6 seconds. Using the equations for constant acceleration and ignoring air resistance we have

y = y_0 + v_{y0}t + \frac{1}{2}g_P t^2

y= \frac{1}{2}g_P t^2 = \frac{1}{2}(7.8m/s^2)(6s)^2 = 140m

His velocity at that time would be v = at = (7.85 m/s^2)(6s) = 47 m/s or about 170 km/h. In comparison, on Earth the highest dive into water was done by Oliver Favre in 1987 from a height of 54m. Even with the lower gravity and greater air resistance on Pandora, 141m seems like a very high distance to fall. Air resistance would lower the jumpers speed by a considerable amount if one gets close to the terminal velocity. On Earth the terminal velocity of a person falling in air is around 56m/s so we might expect that the terminal velocity on Pandora would be about 45 m/s meaning air resistance can not be neglected.