跟讀練習: Vectors and 2D Motion: Crash Course Physics #4 - 通過影片學習英語口語
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128 句
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So far, we've spent a lot of time predicting movement.
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Where things are, where they're going, and how quickly they're going to get there.
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But there's something missing, and something that has a lot to do with Harry Styles.
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And today, we're going to address that.
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We've been talking about what happens when you do things like throw balls up in the air, or drive a car down a straight road.
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That kind of motion is pretty simple, because there's only one axis involved.
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The car's accelerating either forward or backward, the ball's moving up or down, there's no messy second dimension to contend with.
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But this is physics.
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We may simplify calculations a lot of the time, but we still want to describe the real world as best we can.
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And in real life, when you need more than one direction, you turn to vectors.
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Let's say we have a pitching machine, like you'd use for baseball practice.
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We're going to use it a lot in this episode, so we might as well get familiar with how it works.
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We can feed the machine a bunch of baseballs and have it spit them out at any speed we want, up to 50 meters per second.
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The pitching height is adjustable, and we can rotate it vertically so the ball can be launched at any angle.
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It also has a random setting, where the machine picks the speed, height, or angle of the ball all on its own.
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Suddenly, we have way more options than just throwing a ball straight up in the air.
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And now the ball can have both horizontal and vertical qualities at the same time.
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Before, we were able to use constant acceleration equations to describe vertical or horizontal motion, but we've never used it for both at once.
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And we're not going to do that today, either.
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Instead, we're going to split the ball's motion into two parts.
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We'll talk about what's happening horizontally and vertically, but completely separately.
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And we'll do that with the help of vectors.
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Vectors are kind of like ordinary numbers, which are also known as scalars, because they have a magnitude which tells you how big they are.
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But vectors have another characteristic, too – direction.
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Previously, we might have said that a ball's velocity was 5 meters per second.
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And assuming we'd pick downward to be the positive direction, we'd know that the ball was falling down, since its velocity was positive.
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In other words, we were taking direction into account, but we could only describe that direction using a positive or a negative, so we were limited to two directions along one axis.
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But vectors change all that.
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Now, instead of just two directions, we can talk about any direction.
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It might help to think of a vector like an arrow on a treasure map.
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You could draw an arrow that represents 5 kilometers on the map, and that length would be the vector's magnitude.
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But you need to point it in a particular direction to tell people where to find the treasure.
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Which is actually pretty much how physicists graph vectors.
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You take your two usual axes, aim in the vector's direction, and then draw an arrow as long as its magnitude.
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Like, say your pitching machine launches a ball at a 30-degree angle from the horizontal, with a starting velocity of 5 meters per second.
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We can just draw that as a vector with a magnitude of 5 in a direction of 30 degrees.
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Let's say your catcher didn't catch the ball properly and dropped it.
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Then, just before it hits the ground, its velocity might have had a magnitude of 3 meters a second and a direction of 270 degrees, which we can draw like this.
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That's why vectors are so useful.
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You can describe any direction you want.
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But there's a problem.
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One you might have already noticed.
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You can't just add or multiply these vectors the same way you would at ordinary numbers, because they aren't ordinary numbers.
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To do that, we have to describe vectors differently.
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When you draw a vector, it's a lot like the hypotenuse of a right-angled triangle.
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The vector's magnitude tells you the length of the hypotenuse, and you can use its angle to draw the rest of the triangle.
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Right-angled triangles are cool like that.
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You only need to know a couple of things about one, like the length of a side, and the degrees in an angle, to draw the rest of it.
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It's all just trigonometry, connecting sides and angles through sines and cosines.
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Which is why you can also describe a vector just by writing the lengths of those two other sides.
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But those sides are so good at describing a vector that physicists call them its components.
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So let's go back to our pitching machine example for a minute.
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We said that the vector for the ball's starting velocity had a magnitude of 5
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and a direction of 30 degrees above the horizontal.
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We can draw that out like this.
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That's all we need to do the trig.
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The length of that horizontal side, or component, must be 5 cos 30, which is 4.33.
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The same math works for the vertical side, just with the sine instead of the cosine.
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So we know that the length of the vertical side is just 5 sine 30, which works out to be 2.5.
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So our vector has a horizontal component of 4.33 and a vertical component of 2.5.
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In what's known as unit vector notation, we'd describe this vector as v equals 4.33i plus 2.5j.
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The arrow on top of the v tells you it's a vector, and the little hats on top of the i and j tell you that they're the unit vectors,
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and they denote the direction for each vector i just means it's the direction of what we'd normally call the x-axis, and j is the y-axis.
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You'll sometimes see another one, k, which represents the z-axis.
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And, if we wanted to add or subtract two vectors, that's easy enough.
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We just separate them each into their component parts and add or subtract each component separately.
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So 2i plus 3j added to 5i plus 6j would just be 7i plus 9j.
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And minus 2i plus 3j added to 5i minus 6j would be 3i minus 3j.
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Multiplying by a scalar isn't a big deal either.
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You just multiply the number by each component.
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So 2i plus 3j times 3 would be 6i plus 9j.
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The unit vector notation itself actually takes advantage of this kind of multiplication.
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i, j, and k are all called unit vectors because they're vectors that are exactly one unit long, each pointing in the direction of a different axis.
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So when you write 2i, for example, you're just saying, take the unit vector i and make it twice as long.
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But it's not the same as multiplying a vector by another vector.
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That's a topic for another episode.
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So now we know that a vector has two parts, a magnitude and a direction, and that it often helps to describe it in terms of its components.
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And when you separate a vector into its components, they really are completely separate.
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In other words, changing a horizontal vector won't affect its vertical component, and vice-versa.
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And we can test this idea pretty easily.
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Let's say you have two baseballs, and you let go of them at the same time, from the same height.
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But you toss ball A in such a way that it ends up with some starting vertical velocity.
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With ball B, it's just dropped.
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In this case, ball A will hit the ground first, because you gave it a head start.
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Now what happens if you repeat the experiment, but this time you give ball A some horizontal velocity and just drop ball B straight down?
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Which ball hits the ground first?
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It's kind of a trick question, because they actually land at the same time.
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It doesn't matter how much starting horizontal velocity you give ball A, it doesn't reach the ground any more quickly,
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because its horizontal motion vector has nothing to do with its vertical motion.
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With this in mind, let's go back to the pitching machine, which will set up so it's pitching balls horizontally exactly a meter above the ground.
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Then we get out of the way and launch a ball, assuming that up and right each are positive.
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How do we figure out how long it takes to hit the ground?
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That's easy enough.
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We just completely ignore the horizontal component and use the kinetic equations the same way we've been using them.
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In this case, the one we want is what we've been calling the displacement curve equation.
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It's this one.
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We just added y subscripts to velocity and acceleration, since we're specifically talking about those qualities in the vertical direction.
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Now we can start plugging in the numbers.
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The ball's displacement on the left side of the equation is just minus one meter.
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There's no starting vertical velocity, since the machine is pointing sideways, and the vertical acceleration is just the force of gravity.
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Now all we have to do is solve for time t, and we learn that the ball took 0.452 seconds to hit the ground.
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Its horizontal motion didn't affect its vertical motion in any way.
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But sometimes things get a little more complicated.
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Like, what about those pitches we were launching with a starting velocity of 5 meters per second, but at an angle of 30 degrees?
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While we can still talk about the ball's vertical and horizontal motion separately, we just have to separate that velocity vector into its components.
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Just like we did earlier, we can use trigonometry to get a starting horizontal velocity of 4.33 meters per second
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and a starting vertical velocity of 2.5 meters per second.
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Now we're equipped to answer all kinds of questions about the ball's horizontal or vertical motion.
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Here's one.
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How long did it take for the ball to reach its highest point?
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We already know something important about this mysterious maximum.
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At that final point, the ball's vertical velocity had to be zero.
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That's because of something we've talked about before.
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When you reverse directions, your velocity has to hit zero, at least for that one moment, before you head back the other way.
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So in this case, we know that the ball's starting vertical velocity was 2.5 meters a second.
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And we know that its final vertical velocity at that high point was zero meters a second.
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Finally, we know that its vertical acceleration came from the force of gravity, so it was minus 9.81 meters per second squared, since up is positive.
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And we're looking for time, t.
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Fortunately, you know that there's a kinematic equation that fits this scenario perfectly – the definition of acceleration.
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By plugging in these numbers, we find that it took the ball 0.255 seconds to hit that maximum height.
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So describing motion in more than one dimension isn't really all that different or complicated.
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You just have to use the power of triangles.
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In this episode, you learned about vectors, how to resolve them into components, and how to add and subtract those components.
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We also talked about how to use the kinematic equations to describe motion in each dimension separately.
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Crash Course Physics is produced in association with PBS Digital Studios.
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You can head over to their channel to check out amazing shows like The Art Assignment, chatterbox and blank on blank.
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This episode of Crash Course was filmed in the Dr. Cheryl C.
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Kinney Crash Course Studio with the help of these amazing people and our graphics team is Thought Cafe.
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為什麼用這個影片練習口語？
想要提升英語口語，「英語影子跟讀」是超有效的方法！這個影片用生動的物理知識講解，語速適中、邏輯清晰，特別適合用來做「看影片學英語」的素材。影片裡既有日常表達，也有專業術語，能幫你在練習口語的同時，積累多元詞彙。跟著影片進行「shadow speech」，不僅能鍛鍊聽力和反應速度，還能模仿自然的語調，讓你的口語更流暢。趕緊試試吧，堅持下去你會發現自己的口語進步神速！
語境中的語法與表達
影片裡有不少實用的語法和表達，一起看看吧：
- “We've been talking about...”：現在完成進行時，表示從過去持續到現在的動作。比如可以說“We've been practicing shadow speech for a week.”（我們已經練習影子跟讀一週了）。
- “It might help to think of... like...”：用來做類比，解釋抽象概念。例如“It might help to think of vectors like arrows on a map.”（把向量想像成地圖上的箭頭可能會有幫助）。
- “Instead of... we can...”：表示替代關係。可以說“Instead of just reading, we can do shadowspeaks to improve speaking.”（不要只是閱讀，我們可以通過影子跟讀來提升口語）。
發音易錯點提醒
影片裡有些單詞的發音容易出錯，要注意哦：
- vectors：發音是/ˈvektəz/，不要把“vec”讀成“v ek”，而是“v e k”。
- magnitude：重音在第一個音節/ˈmæɡnɪtjuːd/，不要讀成“mag-NI-tude”。
- trigonometry：這個詞比較長，發音是/ˌtrɪɡəˈnɒmətri/，可以分成“tri-go-nom-e-try”來記。
跟著影片多練幾遍，這些發音陷阱就能輕鬆突破啦！
影片中的語法
說話人最常用的結構，並附上影片中的原話：
| 結構 | 影片中的用法 |
|---|---|
| 現在完成進行時 have/has been + -ing — 以前開始、現在仍在繼續的動作 | We've been talking · we've been using · we've been calling |
| 被動語態 be + 過去分詞 — 強調發生了什麼，而不是誰做的 | can be launched · are also known · is produced |
| 現在完成時 have/has + 過去分詞 — 過去發生但與現在仍有關聯的事 | we've spent · we've talked |
| 定語從句 who / which + 從句 — 補充說明人或事物 | numbers, which are · machine, which will |
什麼是跟讀法？
跟讀法 (Shadowing) 是一種有科學依據的語言學習技巧，最初開發用於專業口譯員的培訓，並由多語言者Alexander Arguelles博士普及。這個方法簡單而強大：您在聽英語母語原聲的同時立即大聲重複——就像是一個延遲1-2秒緊跟說話者的影子。與被動聽力或語法練習不同，跟讀法強迫您的大腦和口腔肌肉同時處理並模模擬實的講話模式。研究表明它能顯著提高發音準確性，語調，節奏，連讀，聽力理解和口語流利度——使其成為雅思口語備考和真實英語交流最有效的方法之一。