Tomorrow, When the War Began is a 2010 Australian movie based on the first novel in the epic series by Australian author John Marsden. A fictional “foreign power” invades the country and a band of teenagers have to deal with the situation, eventually learning how to fight back.
Math Scenario 1: Hiding Out
The band of teenagers move between houses on the outskirts of their rural town. The invading forces would be regularly checking the houses for survivors and so the teenagers would need to move regularly.
If the teenagers stay at a house, the chances of them being discovered on any one night is 5%. They decide as a group that they can’t accept a higher risk than 30% of being found at a house. We can calculate the maximum period of time they can stay at the house:
Chances of being safe = (1 – 0.05)^(#nights)
We can set the risk of being found to the maximum allowable amount, 30% or 0.3, which means the chances of being safe are 70%:
0.7 = (1 – 0.05)^(#nights)
This equation can be solved for the #nights by using logarithms. First we can take the log of both sides:
0.7 = (1 – 0.05)^(#nights)
log(0.7) = log[(1 – 0.05)^(#nights)]
log(0.7) = #nights * log(1 – 0.05))
#nights = log(0.7) / log(1 – 0.05))
#nights = 6.95
So the teens shouldn’t spend more than 6 nights at the house before switching to a different property.
Math Scenario 2: Invading the Town in a Garbage Truck
Halfway through the movie the teenagers take command of a large garbage truck and are pursued by machine-gun-armed buggies. The heavy metal garbage truck is reasonably toughened against gunfire but the bullets still take a toll on the vehicle, slowing it down gradually by taking out tyres and doing other damage.
If every minute the soldiers get a critical hit in that slows the vehicle by 10% and the truck starts at 60 km/hr, we can calculate whether they will be able to make it to the safety of the countryside outside the soldiers’ perimeter, 7 km away. The truck will stop regardless after 30 minutes due to a fuel leak.
In 1 minute travelling at 60 km/hr, the truck will travel exactly 1 km:
Distance travelled = 1 km + 0.9 * 1 km + 0.9 * 0.9 * 1 km + …
This type of problem is known as a mathematical series, more specifically a geometric series, because each subsequent term is the result of multiplying the previous term by a constant amount – in this case by 0.9.
The sum of a geometric series is:
where n is the number of terms in the series, and r is the ratio by which each term is multipled to get the next one, and a is value of the first term
So for this situation:
distance travelled = 1 * (1 – 0.9^30) / (1 – 0.9)
distance travelled = 9.576km
9.576 km > 7 km
Looks like the truck will hold out just long enough to make it to safety.
Math Scenario 3: Taking Out The Bridge
At the climax of the movie, the teens attempt to take out a critical bridge by blowing up an oil tanker. They set a fuse to blow up the tanker but have to sprint to make it away safely. If the dangerous explosion radius is 300 metres, and the fuse burns at 30 cm/s, how long a fuse should they set? Assume they are fit teenagers who can sprint at about 8 m/s.
Time to escape = distance to sprint / sprint speed
Time to escape = 300 / 8
Time to escape = 37.5 seconds
If the fuse burns at 30 cm/s:
Fuse length = fuse burn rate * time required
Fuse length = 30 cm/s * 37.5 s
Fuse length = 1125 cm
Fuse length = 11.25 m
Blowing up the bridge doesn’t completely stop the invaders, but it will force them to take a major detour to cross the river safely. If the average speed of the invader convoy is 40 km/hr, how long will it take them to detour the 300 km downstream to the next bridge?
Detour time = detour distance / speed
Detour time = 300 km / 40 km/hr
Detour time = 7.5 hrs
Real Life Example – Counter-IED Equipment
The use of a heavy, metal garbage truck as something that can stand up to a lot of punishment isn’t just something seen in a movie. When cleaning out land mines in wartorn countries, heavy construction-type equipment is modified as it can take a lot of damage if the mine explodes, protecting the operator (or they are remote controlled).
See this example from Fortune 100 company Caterpillar: