In this 2012 movie, aliens come to earth and things get violent. It’s up to a rag tag group of survivors to fight back and save the earth from being conquered by the alien invaders.
Math Situation 1: Sudden Brake
In the final battle of the movie, the old USS battleship Missouri comes out of retirement to fight the alien mothership.
One of the tricks the humans play is to steam in front of the alien ship, wait for the alien ship to fire its projectiles, and then drops its anchor, sudden stopping the ship.
The aliens anticipate where the battleship will be when their projectiles arrive (note the human ship in the picture below is a modern destroyer / frigate but should give the general idea):
So suddenly stopping the ship throws off their aim.
First let’s work out when the aliens should fire based on some distance and timing information.
The human battleship is closing at 30 knots, the projectiles from the alien ship have an average horizontal speed of 300 km/hr, and the alien mother ship is 2000 metres from the nearest point of the battleship’s future path.
First we can calculate how long it will take the alien projectiles to cross the 2000 metre distance. Let’s first convert 300 km/hr into metres/second to make the units consistent:
300 km/hr = 300000 m/hr = 300000 / 3600 m/s = 83.33 m/s
Alien project flight time = flight distance / speed
Alien project flight time = 2000 m / 83.33 m/s
Alien project flight time = 24 seconds
So the aliens should fire 24 seconds before the battleship reaches the position directly in front of it.
Next we can calculate how far back from the final position the battleship will be when the aliens should fire. To do this, we can first convert the speed of the human battleship from knots into metres/second. A knot is 1.852 km/hr.
1 knot = 1.852 km/hr = 1852 m/hr = 1852/3600 m/s = 0.5144 m/s
30 knots = 15.43 m/s
Distance battleship covers in 24 seconds = speed * time duration
Distance battleship covers in 24 seconds = 15.43 * 24
Distance battleship covers in 24 seconds = 370 metres
So the alien mothership should fire when the battleship is 370 metres short of the final position in front of it.
Math Situation 2: F18s Save the Day
After the alien mothership has been partially disabled, the protective shield stopping the other human forces from helping out goes down.
Australian F/A-18 fighters scramble from the aircraft carrier. Given a top speed of about 1900 km/hr, how long will it take them to cross the 500 km from their carrier to the scene of the battle to help out?
Time taken = distance / speed
Time taken = 500 km / 1900 km/hr
Time taken = 0.2632 hrs
Time taken = 15 minutes 47 seconds
Real Life Example – Stopping Distance
Large transport vehicles like road train trucks, ships and trains have long stopping distances that are critical when considering safety.
A large 100000 tonne container ship travelling at 40 km/hr is able to emergency stop and decelerate at a rate of 400 km/hr/hr. How long will it take (both in distance and time) to come to a complete stop?
Time to stop = current speed / deceleration rate
Time to stop = 40 km/hr / 400 km/hr/hr
Time to stop = 0.1 hr
Now we can calculate the distance it will travel during this stopping time:
Especially large supertankers can take much longer to stop – a serious safety concern if there are critical things in their way! On the other hand, some cruise ships can stop much more quickly – a critical requirement given their regular operation in city ports where there is a lot of things that can be damaged if they can’t stop in time.