I have finally decided to write the solution for my previous pop quizz which I felt so lazy to write. Although you have to understand, writing math equations is really annoying. And I didn’t practice LaTex for a long time, so I felt apprehensive about it. Plus, I will have to make more diagrams, which is even more annoying. But still, I like physics and math, and I like writing about them. So, before you get to the solution, firstly, if you know high school physics or above, this could be cake for you, but if you want to proceed, you should see the problem in the link I put above first. Note, you may see that in the movie clip, the solution is already there, but I want to clarify it. Have you reviewed it? Good, now here goes the solution:
As you know from going to the link, poor detective Conan and his friends are trying to escape a building which is about to blow up, and their only hope is to drive a car towards another building which is 60 meters away, 20 meters below them. Now, the question is, is 60 km/hr, or 16.7m/sec enough to reach the building? I am sorry to say, the answer is no. You are going to die, and it will be painful (duh).
In order to understand how to solve this problem, you have to understand decomposition. It means that if an object is moving into 2 directions at the same time, say in a diagonal, because it is being moved by 2 forces at the same time, you can predict where it is going to be by separating the two forces, thereby separating the movement into 2 directions. Then, you can easily calculate where each direction goes to. Then, after getting two separtate positions, just start from a point, and move in one direction first, and then the other, like a treasure map which tells you how many steps to take to the left, and then how many to go up.
For example, say that you are playing megaman, and you lost a bazillion times. Suddenly, you lose your cool, start getting really mad, and start throwing the controller out of the window. In order to calculate its path, you first have to separate its path into up and down, and left and right. The path for up and down is shown in the top picture below. Basically, the force from your hand makes it go up until it stops thanks to gravity and falls down. As you can see, timewise, it makes an arc, but if you map it into one dimenstion, it goes up up up… and then down… down down. You do the same with the forward dimension, map it into the bottom axis, and what do you get? A parabolic curve, as shown in the bottom figure.
That is precisely what we will do with Conan’s cool car. By the way, a nice illustration of this principle is the experiment at the end of the following lecture, for explanation start at 30-35 minute mark, if you want to skip, go to the 40-45 minute mark:
Okay, now, for the car, two forces are acting on it. The forward propulsion and the gravity which sends it down. Those are the two general directions the car is going, which I put it as dotted arrows below. But the way down is different to the way it goes forward, which marches on in a constant velocity. Gravity is a force, which means the car is being accelerated, which means it goes faster and faster and faster every second. I showed this in the figure below as the arrows pointing down getting larger and larger. So the equation for the downward movement requires something a tad bit more complicated, a quadratic one.
The forward movement can be given in the following equation:
The v is velocity, t is time, and x is the distance it traveled. In this case, v is 16.7 m/sec. For now it doesn’t matter though. We know the car will reach the building, but since even if we use that equation to find how long it takes to reach the other building, we don’t know whether it will be below the roof by the time the car reaches it. By then, everyone will die with shards of office windows stuck to them. So instead, I am going to solve this by figuring out how long it takes for the car to fall 20 meters. For this, we need the following equation:
That equation describes what happens to an object as it falls. y is the height, the negative suggests it is going downwards, g=9.8 is the gravitational constant on the surface of the Earth, and t is time. We are trying to solve for time here though, the maximum amount of time needed to reach the roof until it is too late and everyone falls down. So, we multiply by 2 for both sides, divide by g, and apply a square root and we get:
Okay, so it takes around 2 seconds to fall. And if you want to cross 60 meters in 2 seconds, then replace it in the first equation:
Man… It takes at least around 30 m/sec, or 108 km/sec to reach the roof. Sorry Conan, you are dead, and you will blow up in the building with the rest of your friends. But of course, Conan is the main character of the story, so he can’t die, right? I mean, duh, or the story wouldn’t exist anymore. If you are curious about what happens to him, just go to my previous post via the link above and watch the rest of the clip. Frankly, I am not in the mood of explaining any more stuff. ^_^
Now, I would like to think that newbies in physics could learn something new from this post. Of course, I can’t make a good judgment on that because I wrote this. So, what did you think? Did I overexplain some stuff and made it too long? Were my explanations not clear enough? Do I suck so bad that you went permanently blind just by looking at the post? Feedback is encouraged.