How do I survive the elevator shaft in Sector 1 of Metroid: Other M?

I'm in this elevator shaft in Sector 1 in Metroid: Other M. It is after the third navigation station. I climb to the top, killing the enemies along the way, and there are some exposed wires at the very top. When I shoot the exposed wires, the elevator moves a bit and it shows me a bad guy climbing up the shaft. I don't know how to kill him before the elevator falls. At first I tried just shooting missles at him, but I wasn't fast enough and the elevator fell on me. So then I tried hiding in one of the open areas on the right side of the shaft and trying to kill the bad guy. But as soon as he got to where I was he'd grab me with his tail and then the elevator would immediately fall on both of us. Am I missing something?

You cannot kill the bad guy using conventional weapons or combat tactics. Instead, you must use the very elevator itself to eliminate this impending threat.

What you're shooting is a piece of wreckage blocking the elevator from moving. However, it's not the only piece of wreckage. There is a second one on the side from where you shoot the first.

As you have noticed, there's an open platform on the side, where the elevator can't hit you.

However, once you get there, don't try to shoot the enemy coming up the shaft! Instead, look to the other side!

Fire your missiles to destroy this piece of wreckage, and to send the elevator crashing down on your enemy.