an interminable conversation 9: some basic physics
yer most basic sine graph
Jacinta: So it’s time for us oldies to go to school, and get into physics from scratch, including the maths.
Canto: Yes, we’re not going to go all historical this time, much as we love all the nerdy characters to be encountered, instead we’re going to go with the concepts, from simple to complicated. I’ve found a collection of videos, called ‘crash course physics’, and we’re going to follow the ineffable logic of the presenter, Dr Shini Somara, to reach the pinnacle of sagesse en physique. Starting with basic motion in a straight line.
Jacinta: Exciting. I’ve done that. But in this first episode she deals with cars and acceleration, inter alia, including its maths. Equations! Time, position, velocity and acceleration will be explained/analysed in simple terms for this starter.
Canto: Kinematic equations – we’re going to the Kinema! So, motion in one direction on a straight line. You’re stopped at a red light, and then put your foot on the accelerator when it goes green. Seven seconds later (precisely), there’s a siren behind you – a police car is asking you to stop. They give you a ticket for speeding in a 100kph zone.
Jacinta: So, in 7 seconds you’re up to more than 100kph? I know nothing about cars but that’s – unusual? Is it?
Canto: I’m sure car nerds can tell us, but so can google. There are plenty of cars that can get to 100 in less than 4 secs, even less than 3. Supposedly. Anyway, you’re doubtful about the police claim, but you can’t be sure, your speedometer is stuffed. How can you challenge the police claim, using maths?
Jacinta: You can’t, and anyway in Australia you’d be defected for a stuffed speedo.
Canto: But this is the USA, the land of shitty libertarian laws. So you’re travelling in one direction, one-dimensionally, so to speak. So the key variables here are the afore-mentioned time, position, velocity and acceleration. We also have to bear in mind change in position, aka displacement, which could have a positive or negative value – in this example, clearly positive. Now, velocity is about how that displacement occurs over time. It also can have a positive or negative value. Acceleration is about changes in velocity over time. You can feel that change – positive or negative – when you’re ‘thrown’ forward or backward on acceleration or braking.
Jacinta: So Dr Somara presents graphs that are fairly easy to read for a stationary vehicle, and one moving at a constant velocity. The vertical x-axis measures position or displacement in metres from an initial position, the y axis measures time. A stationary vehicle will show a straight horizontal line from the moment it stopped until it starts to move again. Constant velocity will show a straight line moving diagonally along both axes. An accelerating vehicle will of course show a curving line, curving up to the vertical, while a decelerating one will be curving to the horizontal.
Canto: So that’s a simple position v time graph, now to look at velocity and acceleration slightly differently, with velocity in metres/second on the vertical axis and time on the horizontal, and with acceleration in metres per second per second, that is, metres per second squared, on the vertical axis, and time, in seconds, on the horizontal. So this relates all our variables, time, position, velocity and acceleration. Average velocity is the change in position over time, and acceleration is the change in velocity over time. To get average velocity you divide change in position by change in time.
Jacinta: But as Dr Somara says, subtraction is also a feature – to find out ‘change’ you subtract initial value from final value, which sounds right but somehow seems to contradict the previous….
Canto: One’s talking about a change, the other about an average. They’re quite different. So the change in a particular value, or variable, is symbolised or abbreviated as delta, ∆. So, v = ∆x/∆t, average velocity (the v should have a bar above it, but I haven’t learned how to do that – will I need an extra keyboard?) equals change in position over change in time. For Dr Somari’s example, the car moved from 4 metres to 13 metres (the change in position), i.e. a value of 9 metres for ∆x. This occurred over 3 seconds, apparently, which divides as 3m/sec for average velocity over that period. But of course the car was accelerating during that period. The equation for acceleration is a = ∆v/∆t, for average acceleration.
Jacinta: Okay, and we can, apparently handily, rearrange the equation to get v(average) = v(at time zero) + at. This equation is called the Definition of Acceleration. Tadaaa! Constant acceleration is equal to the change in velocity divided by change in time. This is the first of the two main kinematic equations, which links velocity acceleration and time.
Canto: Okay now our physicist turns to gravity (g), which here on Earth is a force causing acceleration at 9.81 m/sec squared. But then she talks about the second kinematic equation, the Displacement Curve, which involves acceleration, starting velocity and time in order to calculate displacement:
x(position) – x(at time zero, initial position) = v(initial velocity)t +1/2a(acceleration)t(squared).
All of which looks very messy because I haven’t learned how to do the proper notation. Anyway this links acceleration as change in velocity to velocity as change in position. Right?
Jacinta: Uhhh, yeah. And the other kinematic equations, we’re assured, are just rearrangements of this dynamic duo. So apparently this takes us back to our speeding issue at the start. The initial velocity was 0, the time was 7 seconds. The displacement curve²equation/formula can be used to work it all out, or at least the acceleration. Our physicist tells that x – x (initial position) is 122 metres, which equals initial velocity (zero) multiplied by 7 seconds (which must surely be zero?) plus I/2a (which is to be found) multiplied by 7s squared, which is 49 seconds. So 122m = 0 + 49 (49) multiplied by half the acceleration, which by calculation I discovered to be close to 2.5, so the acceleration was approximately 5 metres per second squared.
Canto: It works out! And, following our expert, we can use the Definition of Acceleration formula to arrive at final velocity. It’s basically V + at, or 0 + 5 X 7, so a speed of 35 m/sec, which in km terms is about 126 km/h. Amazing! We got the maths. There is hope!
Jacinta: Well they’re diving into the deep end with crash course physics, as the next video is all about calculus and derivatives. About which I have no idea.
Canto: Yes, maths are the basis of physics, and we lost contact with complex maths decades, though I’m quite good at multiplication. But calculus, duh. Though our teacher tells us that it’s all just about accurately describing change.
Jacinta: Important – she goes on to explain things called derivatives, but I note in the inset:
Not all equations have derivatives! When we say ‘equations’ here, we really mean a function – an equation with only one output for each input. More specifically, we’re talking about functions that have derivatives.
I’m looking forward to clarification of all that.
Canto: So calculus explains the why’s and wherefores of change through derivatives. She also mentions integrals early on, as ways of calculating area under a curve – which we actually mentioned in those terms in a previous post.
Jacinta: We sound smart sometimes. So, derivatives. Dr Somara returns to the car and speeding example. The car drives off after the police incident, accelerating of course. But we don’t have a direct measure of the acceleration, but we know positional change over time. This is apparently equal to amount of time driving, squared, X = t². After 20 secs of driving, some kind of roadside ‘detector’ reveals the car’s speed. The driver takes time to register that she’s going even faster than 126 mph.
Canto: Dumb blond? Maybe not, maybe the detector is dodgy. How to find the velocity at the moment she passes it? Which, according to Dr Somara is the derivative of her change in position. And this is also about limits. These are key ideas:
Limits are based on the idea that if you have an equation on a graph, you can often predict what it’s going to look like at one point, just by knowing what it looks like at the surrounding points.
Jacinta: So our teacher gives the example of graphing x = t² when t approaches the limit of 0. So remember we have our time on the horizontal, and distance covered (or displacement, or positional change – it seems ‘distance’ is a no-no in this maths) on the vertical axis. So, moving back to zero from t=1 and x=1 she finds that when t=0.5, x=0.25, and when t reaches 0.1, x=0.001, so both values approach zero. This apparently shows what happens when you make intervals smaller. Another definition:
An interval is just a range on a graph. It’s the space between two points on the horizontal axis.
Of course, because that’s the time axis, generally. This is great parroting, but then when parrots copy their trainer perfectly they’re regarded as brilliant.
Canto: So we’re calculating the average velocity over a particular interval – from 15 to 20 secs. We use the equation v = ∆x/∆t (∆x is change in position, ∆t is change in time). The change in position, after subtraction, was 175 metres, the change in time 5 secs. So the average velocity works out as 35 m/sec. But this is only an average, and doesn’t take into account acceleration. But using limits gets us closer to the number we want. You can calculate your average over increasingly small intervals to arrive at an increasingly accurate figure.
Jacinta: So, sticking with our teacher, velocity is an equation that describes change in position, acceleration describes change in velocity. Velocity is thus the derivative of position and acceleration is the derivative of velocity. This is expressed in writing, using, for example, the power rule, expressed using variables and their numbered exponents. So x = t² is an equation that works here. To calculate the derivative, you take the exponent, 2, and put it in front of the variable, and subtract 1 from the exponent, and that’s the derivative, 2t. In full, the derivative of x = t² is 2t.
Canto: That’s a trick, as Dr Somara said, but it’s not really explained. She says ‘no matter how [you’re accelerating], your velocity will be 2t – double the number of seconds’. So I think it depends on those seconds. After 5 seconds, say, you’re travelling at 5m/sec, but after 20 secs, your speed is 40m/sec. So dx/dt = 2t ‘which is just a way of saying, mathematically, we’re taking the derivative of x with respect to t’. But it’s also written differently sometimes: if f(t) = t², then f'(t) = 2t. And I’m guessing that f stands for function, but I don’t quite know what a function is.
Jacinta: A function is:
in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). Functions are ubiquitous in mathematics and are essential for formulating physical relationships in the sciences.
That’s from Britannica online. So to continue, if f(t) = t², then f'(t) = 2t. That’s to say, f prime (t) = 2t, according to our teacher, who doesn’t explain ‘prime’. Do we have to do a maths course before we do this physics course? Does it have to do with prime numbers?
Canto: Apparently not. The symbol can serve a number of purposes in maths. Let’s just leave it for now. Using the power rule we can find other derivatives, e.g. x = 7t to-the-power-6. This equation has the variable t, and its exponent 6. We take the exponent and put it in front of the 7t variable, multiplying the number and subtracting 1 from the exponent, 42t to-the power 5. That’s to say dx/dt = 42(t to the power of 5). But maybe that shouldn’t be bracketed. And when the exponent is a fraction or decimal, the derivative of, say t to the power of one half is 1/2t to the negative one half. You always minus one, I don’t know why.
Jacinta: Ours is clearly not to reason why, at least not yet. This derivative trick works for negatives too. In the case of x = t to-minus-2, the derivative (dx/dt) = -2t to minus 3. Not very comprehensible, and then she mentions the dread word, trigonometry, used for calculating triangles, their angles and sides. Apparently physics uses right-angled triangles a lot. We shall see.
Canto: Indeed, let’s get into it. The derivatives of sine x and cosine x. If you have a right-angled triangle with an adjacent angle x, sin(x) = the length of the opposite side divided by the hypotenuse. For cosine, cos(x) it’s the length of the adjacent side divided by the hypotenuse. So, sin(x) = o/h, cos(x) = a/h. ‘So the graphs will tell you what those ratios will be, depending on the angle’.
Jacinta: I’m not sure if I really understand this, but let’s move on into further weird territory, in which sin(x) is plotted on a graph going from -360° to 360° on the x (horizontal) axis (that’s the ‘phase’, in degrees) and -1 to 1 on the y axis. At x = -90° and x = 90° the curve turns – that’s at every 180°. At those points the equations aren’t changing and the derivative is zero. Between the points the derivative oscillates from positive to negative. That derivative is in fact cos(x). I’m not sure why, but the derivative of cos(x) is -sin(x), the derivative of -sin(x) is -cos(x) and the derivative of -cos(x) is sin(x), for future reference. I’m hoping it’ll all become clear some day. Graphing all these will provide the proofs, evidently.
Canto: Yes, so Dr Somara finishes off this vid with another derivative that’s important in calculus, e×, the derivative of which is also e×, always. e, like π, is an irrational number which is quite vital to calculus, apparently. And even finance. Can’t wait to find out. So with the preceding we can, supposedly, take any equation for position and calculate the derivative, and so, velocity. And for velocity, your acceleration. Using integrals, which we’ll soon learn about, we can go backwards from acceleration to velocity, and from velocity to position. Presumably that will be next time.
Jacinta: So easy…
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