Physics with Portals

For some reason, one physics problem that has stuck with me since I was an undergrad is the “shoot the coconut” problem. The scenario involves a coconut falling from a tree and another projectile being thrown or launched at it. Students have to calculate the right angle and velocity of the projectile to hit the coconut before it hits the ground.

The Puzzle Maker is perfect for bringing physics problems to life. Here’s my take on shooting coconuts in Portal 2.

Let’s do a little bit of informal physics to calculate the velocity of cube 2 when launched. For simplicity’s sake, we’ll label the falling cube C1, and the launched cube C2.

C1 falls about 13 panels (units) before striking C2. Using the equation for displacement, we find that it takes about

t = sqrt (2*h/g)          (1)

to reach the point of impact. By equation 1, t = 2.35 s. That is, of course, without air resistance. Taking into account air resistance and the amount of time it takes for the cube to actually leave its dispenser give us t = 3.6 s (calculated with a stopwatch, by the way).

It takes almost 2.7 s for C2 to drop, roll, hit the AFP and get launched through the air (this number actually varies from test to test because cubes don’t always leave their dispensers in exactly the same way).

That leaves us with about 1.1 s for C2 to leave the angled panel, fly through the air and strike C1 as it falls. C2 leaves the orange portal at an angle, θ = 45 degrees and has to travel 4 panels in the x-direction to reach C1. So, in the x-direction, C2 has a velocity, v_x = 3.6 u/s. Using trigonometry, we start with

v_x = v*cos(θ)          (2)

and then rearrange to find

v = v_x/cos(θ)          (3)

which tells us the initial velocity of C2. After plugging in our variables, equation 3 tells us that C2 has an initial velocity of v = 6.9 u/s.

Gotta love physics.