Throwing and Dropping ObjectsDate: 11/26/2002 at 23:24:14 From: Amanda Evans Subject: Math I have tried asking my teachers this question, but they say I am too young to understand the answer. I have even asked my parents, but they give me the same answer. I cannot find this answer anywhere. My question is very simple. Why, when you throw something in one direction and drop something at the same time, do they land on the ground at the same time? Also, when you throw something in the air, does it come down at the same pace as it went up? Thank you for taking the time to read this. Date: 11/27/2002 at 10:59:15 From: Doctor Ian Subject: Re: Math Hi Amanda, It's a good question! A lot of people only understand this kind of thing in terms of equations using variables, so they can't explain it to someone who hasn't studied algebra yet. But let's see if we can skip the equations and just talk about the ideas, okay? The answer to your first question assumes that you throw the item in a horizontal direction. Viewed from the side, the situation would look sort of like this: throw ' ' -----> ' ' d ' ' r ' ' o ' ' p ' ' ' ' ' ' ' If you throw the item in an upward direction, it will take more time to hit the ground. If you throw it in a downward direction, it will take less time. (By the end of this, you should understand why.) It might be easier to see what's going on if we imagine that you and I are going to have an odd kind of race. We're both going to start at the same point, and you're going to move south, following a simple rule: In the first second, you'll take one step; then in the second second you'll take two steps; then three steps; then four steps; and so on: X | 1 step per second --- | 2 steps per second | --- | 3 steps per second | | --- | 4 steps per second | | | --- What this means is that as time goes by, you'll be moving faster and faster. It's like what a car does as it pulls away from a stop sign. In the first second, it goes 5 feet; then 15 feet in the next second; then 35 feet in the next second; and so on. This kind of motion is called 'acceleration', which is just a fancy word for 'changing speed'. We say that the car 'accelerates' from a speed of zero to whatever its final speed (say, 45 miles per hour) turns out to be. (When the car comes to the next stop sign, the speed decreases, and sometimes we call that 'deceleration', but in both cases, it's really an acceleration, that is, a change in speed.) So in our race, you're accelerating toward the south. While you're heading directly south, I'm going to do something a little sillier. In the first second, I'll take one step east and one step south; then I'll take another step east and two steps south; then one and three; and so on. What would that look like? X--- | | 1 1--- | | | | 2 2--- | | | | | | 3 3--- | | | | | | | | 4 4 So at the end of 4 seconds, I've moved to the east at a constant rate, and you haven't; but we've _both_ moved toward the south, and with the same acceleration. If we change 'south' and 'east' to 'down' and 'over', this is more or less what happens when you drop one thing and throw a second thing in a horizontal direction. Note that it doesn't matter how fast I move to the side. If I move three steps east during each second, it looks like this: X--------- | | 1 1--------- | | | | 2 2--------- | | | | | | 3 3--------- | | | | | | | | 4 4 So the curve that I make - or the curve made by the thing you throw - has a flatter shape, but it doesn't affect the southward/downward movement. Let's think about the car again. What makes it accelerate? In short, we get the engine to burn fuel to produce energy, and we use the energy to apply a force to the car - just as if we got out of the car and started pushing on it from behind. (The engine can generate a lot more force than a few people, though!) So whenever we apply a force to something, it accelerates. The more force we apply, the more acceleration we get. (Sometimes in car commercials, you'll hear that a car can go 'from zero to 60 in ___ seconds'. If a car can accelerate quickly, it means that the engine can generate a lot of force.) When you drop a ball, it accelerates too. What makes it accelerate? As we currently understand it, everything in the universe exerts a little bit of force on everything else, and we call this force gravity. So when you drop a ball, the earth is exerting a force on the ball in the same way that an engine exerts a force on a car. One difference is that when the car is going fast enough, we stop pushing on the gas pedal, so it doesn't _keep_ accelerating. But the earth never stops pulling. Even when the ball gets to the ground, the earth doesn't stop pulling. When you step on a scale, and it tells you that you weigh ___ pounds, what it's telling you is how hard the earth is pulling on you. If the floor were suddenly to move out of the way (for example, if you were in an elevator that was standing still, and the cable broke), you'd start accelerating. The reason you _don't_ accelerate is that while the earth is pulling on you, the floor is pushing on you, and the forces exactly cancel out. It's sort of like if four people got out of a car, and two started pushing from the rear, and the other two started pushing from the front. They'd generate a lot of force, but the car wouldn't go anywhere. And so we have to amend our previous statement - whenever we apply a _net_ force to something (that is, a force that isn't being canceled out by some other force), it accelerates. So what is happening when you drop a ball? There is nothing to push upward, so the downward force of gravity starts accelerating the ball towards the earth. And it keeps accelerating until it hits the ground. Then a lot of other things happen that we're not going to talk about right now. :^D And what is happening when you throw a ball to the side? Again, nothing is pushing upward, so gravity starts accelerating the ball toward the earth. The ball increases its speed in the downward direction. What about toward the side? There are no forces acting in that direction, so it just keeps its original speed. So if you throw two balls, one twice as fast as the other, they'll hit the ground at the same time, but one of them will travel twice as far to the side. Dropping a ball, then, is just 'throwing' it with a sideward speed of zero. There is one other important thing to know about gravity. If we put the same engine in two different cars, one much lighter than the other, the lighter one will accelerate more quickly. In each case, the engine generates the same force, but in the lighter car there is less to push. But the force of gravity depends on how much stuff there is to pull on. So let's say you have two balls, one of which weighs twice as much as the other. If you drop them at the same time, from the same height, the earth will exert twice as much force on the heavy one, but there is twice as much weight to be moved. So the extra weight cancels the extra force (sort of as if you bought a Porsche, but put half a ton of iron in the trunk), with the result that the two objects end up with the _same_ acceleration. That is, if you drop a baseball and a cannonball at the same time, from the same height, they'll hit the ground at the same time. In fact, if you go to a place where there isn't any air (like the moon), you can drop a feather and a piano, and they'll hit the ground at the same time. So what happens if you throw a ball upward? Now it's like a car with the brakes on. Gravity is pulling on it, so it decelerates until its speed gets all the way to zero: ^ 0 (stopped) ^ 1 'steps' per second | ^ 2 'steps' per second | | ^ 3 'steps' per second | | | start But then it's just like dropping it from a new height. v 0 | 1 v | 2 | v | 3 | | v start Since the acceleration is the same no matter which way the ball is moving, when the ball gets back to its starting point, the speed coming down is the same as the speed going up. If you're standing at the edge of a cliff, and you throw the ball up at a certain speed, when it gets to the bottom of the cliff it will be moving as fast as if you'd thrown it _down_ with the same speed: ^ ^ | | v ^ | | | | v ^ | | | | | | v --------- | Now it's just as if you threw it downward. | | | | | | | v | | | | | | | | | | |_____v_____ Does this all make sense? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 11/27/2002 at 18:49:27 From: Amanda Evans Subject: Math That all makes sence to me, but now I would like to know the equation for all of this. There must be a formula about my question. Date: 11/27/2002 at 22:23:57 From: Doctor Ian Subject: Re: Math Hi Amanda, In fact, there _is_ a formula for figuring out how far something falls in a given amount of time. Here is an example of its use: Wile E. Coyote Lands in the River http://mathforum.org/library/drmath/view/56333.html And here is a slightly more interesting use: From What Height Did the Stone Fall? http://mathforum.org/library/drmath/view/56372.html And here is a more practical use: Real Life Uses of Quadratic Equations http://mathforum.org/library/drmath/view/60810.html Here is an explanation of where the formula comes from: Projectile Motion http://mathforum.org/library/drmath/view/56348.html It might be a little over your head, but I thought you'd like to see it anyway. I hope this helps. Write back if you'd like to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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