##
**Black Lagoon**

Welcome
class and please get out your physics books as this week’s lesson is on trajectory
and Black Lagoon. No we will not be
examining bullet trajectories from the numerous shootouts found in the show,
but the infamous boat scene. For those
of you who are not familiar with it, the clip can be found below.

###
**Considerations**

Now just
at first glance, there are several factors that need to be examined, to
determine if this stunt is possible in real life. The speed of the boat, angle of the ramp,
height of the helicopter, not to mention the strength of the ramp, and the strength
of the boat’s hull. For the purposes of
this examination I will not be including the strength of the ramp or the
strength boat’s hull. The reason for
this exclusion is because a PT boats hull is made out of wood, specifically
mahogany, which while strong, does have its limits. This is also the reason for excluding the
strength of the sunken boat that is used as a ramp. From the few shots that are shown of the boat
it sports a fair amount of rust, and has been sitting there for an undetermined
length of time. Given the amount of
force needed to launch a boat into the air, it is entirely possible that the PT
boats hull or the ramp could give out as soon as they collide ending the myth
right there. Now given that the ramp is
successfully used in the clip, we will assume that both the hull of the boat
and the ramp are strong enough.

###
**Terminology**

In order
to determine if the Black Lagoon can attack the helicopter with torpedoes, the
trajectory of the Black Lagoon must be determined. Trajectory is the path a projectile takes
through space. While the Black Lagoon is
a PT and not a true projectile, once it leaves the water and can no longer
propel itself, it can be considered one.
The type of trajectory that we are interested in is a ballistic
trajectory. In this type of motion the
object moves both vertically and horizontally near the earth’s surface. The force of gravity is the main force acting
on the projectile so other forces like air resistance are excluded as their
influence in this case would be negligible at best.

###
**Basic Facts of the scene**

Before
we begin diving into the math let’s consider the facts that will be used to
consider if this particular scene is in fact possible. In the picture below we can see that at the
peak of its trajectory the Black Lagoon reaches a height greater than that of
the helicopter.

We can
use a second picture to estimate the approximate altitude of the helicopter.

In this
picture we are concerned about the trees shown in the background as a way to
determine the approximate altitude. We
can assume at this point in the show that the Black Lagoon is in an estuary
close to Ronanapur, a fictional city in Thailand. Mangrove trees are a common tree found in
estuaries in Thailand. There are several
different species mangroves with a range of heights from 6 meters on the shorter
side to species that can grow up to 35 meters on the larger side. Now let’s assume that trees beneath the helicopter
are of the shorter variety at 6 meters.
An examination of the image shows that the helicopter is two tree lengths
from the water at a height of 12 meters. Given the first image the Black Lagoon must
reach a height of at least 12 meters for the scene to be possible.

###
The Math

Projectile
motion is motion in two directions and the equations for each are shown below.

**Glossary**

x- Horizontal position

y- Vertical position

v- velocity- speed

t- Time

a- acceleration- change in speed= to the force of gravity at
9.8m/s^{2}

f- Final

i- Initial

**Determining horizontal position**

X_{f}
= vt + X_{i}

**Determining vertical position**

Y_{f}
= Y_{i} +v_{i}t_{i} +at_{f}^{2}

V_{f}^{2}
= V_{i}^{2} +2a(Y_{f} – Y_{i})

The
above equations work in situations where the object is not projected at an
angle like a plate sliding off the edge of a table. If the projectile is launched at an angle
another wrinkle needs to be added in order to complete the calculations for a
parabolic trajectory shown below.

As shown
in the diagram the angle of the launch determines the height and distance the
projectile travels assuming each one is launched with the same initial speed. For the purposes of this evaluation only the
vertical component of the Black Lagoons trajectory will be considered as it
does fly past the helicopter in the scene, and they can be considered close
enough for the torpedoes to hit the target.
The maximum height will be determined for each of the three angles shown
above

**Calculating the vertical component of the initial velocity**

V_{yi} = V_{xi}(sin angle)

The
maximum speed of a PT boat is 76 kmph and it can be assumed that Dutch keeps the
Black Lagoon in top form. To make the
math easier 76 kmph converts to 21.1 meters per second.

**Vertical speed at 30 degrees**

V_{yi}
= 21.1m/s x sin30

V_{yi}
= 21.1 x 0.5

Initial
vertical velocity at 30 degrees = 10.6 m/s

**Vertical speed at 45 degrees**

V_{yi}
= 21.1m/s x sin45

V_{yi}
= 21.1m/s x 0.71

Initial
vertical velocity at 45 degrees = 14.9 m/s

**Vertical speed at 60 degrees**

V_{yi}
= 21.1m/s x sin60

V_{yi}
= 21.1m/s x 0.87

Initial
vertical velocity at 60 degrees = 18.4 m/s

**Calculating the maximum height of the Black Lagoon**

Maximum height = (V_{yi}^{2}) / (2a)

**At 30 degrees**

X= (10.6)^{2}
/ 2 x 9.8

X = 112.4
/ 19.6

Max
height = 5.7 meters

**At 45 degrees**

X= (14.9)^{2}
/ 2 x 9.8

X = 222
/ 19.6

Max
height = 11.32 meters

**At 60 degrees**

X= (18.4)^{2}
/ 2 x 9.8

X = 338.6
/ 19.6

Max
height = 17.3 meters

###
Conclusion

Well the
numbers don’t lie guys, and to be honest I was surprised by this one. If the angle of the sunken boat the Black
Lagoon uses as a ramp is more than 45 degrees then the boat clears the 12 meter
mark and the stunt works. I did not
expect to be able to call this one confirmed but here it is

The
infamous PT boat vs helicopter scene can actually work if we assume the hull of
the boat and the ramp can survive the stress.
Plus Dutch has to be able to aim the torpedoes to hit the helicopter.

###
**Note**

For
anyone who is reading, if you could let me know what you think on this one since
I am mainly a biology and chemistry teacher.
Plus it’s been a while since I've done any physics. Any suggestions you guys have are welcome. I am always looking for new ideas.