Logan is a 2017 superhero film starring Australian actor Hugh Jackman in his (most likely) last role as the Wolverine. Set in the future, mutants are near extinction and still being hunted for exploitation by the military. Wolverine must try to save the remaining mutants in a series of high octane chases and fights.
You can view the trailer here (note Logan is R-rated so not suitable for children):
Math Scenario 1: Self-driving Trucks
The movie features one of the most scary depictions of autonomous driving in the future, in this case huge autonomous trucks racing down the highway. Although it’s possible some of the trucks in the movie were hacked to cause a certain accident, even their normal behaviour would potentially be very dangerous.
They appear to sound their horn but not slow down particularly if they detect a hazard on the road ahead, and they’re travelling pretty fast too. Perhaps human drivers aren’t meant to be on the road mixing it up with them.
Nevertheless, we can work out, even with a zero second reaction time, how long it might take such a truck to stop if a child jumped out in front of it on the highway.
The stopping distance formula for a vehicle is:
v_initial is the initial speed of the vehicle – let’s use current US highways speeds of 65 mph = 104.61 km/hr = 29.06 m/s
g is gravity
d_braking is the estimated braking distance
f is the friction coefficient. A reasonable coefficient is 0.7, although this depends on weather and road surface conditions
G is the road gradient (the slope of the road) – we can use a flat road with 0 gradient
With reaction time not relevant (let’s imagine the truck’s AI system detect and responds so quickly as to have no effective reaction time, unlike a human driver), we can calculate the pure stopping distance. First we need to re-arrange the formula to be in terms of the braking distance:
v_end^2 = v_initial^2 – 2 * g * d_braking * f
2 * g * d_braking * f = v_initial^2 – v_end^2
d_braking = (v_initial^2 – v_end^2) / (2 * g * f)
Then we can substitute in real values:
d_braking = (v_initial^2 – v_end^2) / (2 * g * f)
d_braking = (29.06^2 – 0^2) / (2 * 9.81 * 0.7)
d_braking = 61.49 m
Math Scenario 2: Escape Across the Border
Near the end of the movie, the mutant kids make a desperate dash to cross the Canadian border on foot before the baddies can catch up.
The baddies first send out a fleet of drones to perform reconnaissance. Once they’ve found the kids, they then send out a convoy of vehicles and people on foot to capture them.
Let’s say the kids start 10 km from the border, and can only move at about 2 kmh/r due to the rough terrain and their young age. They have a 4 hour headstart.
The drones then fly out along the same route at about 20 km/hr, flying slowly so their heat sensitive equipment works optimally.
How far into the flight will the drones overfly the kids?
If the trucks start out at 40 km/hr as soon as the kids are located, will they reach the kids before they cross the border?
First we can work out when the drones overfly the kids, which will be when their journey distance is the same:
d_kids = 2t
d_drones = 20 * (t – 4) for t >= 4
If we equate these two equations:
d_kids = d_drones
2t = 20 * (t – 4) for t >= 4
2t = 2ot – 80
18t = 80
t = 4.444 hrs
So the drones catch up with the kids 4.444 hrs after the kids start out, or only about 27 minutes (0.4444 hours) after the drones start out.
At this point in time we can work out how far the kids have left to go:
kids remaining distance = total distance – distance already covered
kids remaining distance = total distance – speed * time
kids remaining distance = 10 – 2 * 4.444
kids remaining distance = 1.112 km
The time the kids will take to travel this distance is:
remaining kid travel time = distance remaining / speed
remaining kid travel time = 1.112 / 2
remaining kid travel time = 0.556 hours
The trucks have to travel 10 km, but can go faster, at 40 km/hr. Let’s find out how far they take to travel the 10 km to the border:
truck travel time = distance / speed
truck travel time = 10 / 40
truck travel time = 0.25 hrs
While the kids will take 0.556 hours to get the remaining distance to the border, the trucks will be able to get there in only 0.25 hrs.
This means the kids are going to have to stand and fight (what happens in the movie) or come up with another plan.
We can also visualize this scenario with a graph of the positions of the kids, the drones, the drone intercept point, and the position of the convoy over time:
Math Scenario 3: How Old is Logan?
Wolverine looks pretty worse for age in the movie, and part of that is because he’s quite old and his regenerative powers are fading.
The movies and comic books aren’t all consistent with showing how old he is, but we can do some basic calculations.
In Wolverine Origins, a young Wolverine is shown in 1845 first realising his powers as his bone claws come out of his knuckles. If he is a young teenager – say 12, then we can calculate that his age in the Logan movie is:
Age = logan year setting – wolvering origins setting + age in origins
Age = 2029 – 1845 + 12
Age = 196 years old
Real Life Example: Agricultural Robots
Midway through the movie Wolverine, Charles and X23 stop at a farmhouse. Their small farm is surrounded by a large corporate farming operation, which uses huge robotic harvesters.
One of the standard tasks of these mobile farming robots it go up and down rows of a farm, spraying weeds or harvesting crops. They rely on positioning systems like GPS (the Global Positioning System) to guide them along these rows accurately. Other systems use cameras to do vision-based positioning.
There is a lot of mathematics including probability, statistics, geometry and trigonometry involved in making these systems work!