Narrator: Before Alison tackles the final equation, let's look again at the two general rules she's been using. Most importantly, remember that an equation is like a set of scales, so that you can do almost anything you like to an equation, PROVIDED you do the same thing to the other side, to keep the balance. The second rule helps you to decide WHAT it is that you should do to both sides of the equation. So to undo an operation you should do the opposite, to undo a subtraction you add, to undo a multiplication you divide, to undo a square root you square and so on. The final equation can be rearranged using exactly the same principles, remembering to keep going in a series of small steps. It's like peeling away layers of an onion. But it needn't end in tears!

Shelagh: So this is your last equation, looks a bit horrible. The delta, like before, is a change in something ...

Alison: Right.

Shelagh: ... so delta E k is a change in the kinetic energy ...

Alison: Okay.

Shelagh: ... of an object of mass m that increases its speed from u to v.

Alison: Okay.

Shelagh: And what I'd like you to find is an equation for u.

Alison: [Big sigh].

Shelagh: You are.

Alison: [Nervous laugh]

Shelagh: We don't go for the u straight away, we'll go for this whole term here, and let's try and at least get that on the left-hand side.

Alison: Yes.

Shelagh: So what would you have to do to get that on to the left-hand side?

Alison: Um, er, well it's subtracted here ...

Shelagh: Yeah.

Alison: ... so if I add half m u squared to both sides ...

Shelagh: Yes.

Alison: Er, I'll just, so if I do delta E k add a half m u squared, that is equal to a half m v squared minus a half m u squared plus a half m u squared.

Shelagh: That's fine. So you, you've added that to both sides.

Alison: Yeah.

Shelagh: So you've preserved your balance again. And so what can you cancel?

Alison: Those two cancel each other out.

Shelagh: They do.

Alison: So I'm left with, er, ...

Shelagh: That's fine. You're left with ...

Alison: ... delta E k ...

Shelagh: ... E k

Alison: ... plus half m u squared is equal to half m v squared.

Shelagh: Right. So now you've got half M U squared where you want it.

Alison: Yes.

Shelagh: But you've still got it added to delta E K. So what can you do next?

Alison: Subtract that from both sides.

Shelagh: Yes.

Alison: Um, so delta E k plus half m u squared minus delta E k is equal to a half m v squared minus delta E k.

Shelagh: Yes.

Alison: And these two terms cancel each other out.

Shelagh: E k minus E k is zero.

Alison: Yes. So that leaves me with half m u squared is equal to half m v squared minus delta E k.

Shelagh: Good, okay. So now we're really making progress ...

Alison: Yeah.

Shelagh: ...'cause you've only got one term on the left, at least.

Alison: Yeah, yeah.

Shelagh: ... but you've still got half and you've got M in it.

Alison: Yeah.

Shelagh: So what could you do next?

Alison: Well a half m u squared is the same as m u squared over two, divided by two so if I multiplied both sides by two ...

Shelagh: Yes.

Alison: So, I'll just, actually I'll just write this out, if I've got m u squared over two.

Shelagh: Right. That's exactly equivalent to that, yes.

Alison: Yes, yeah. So if I multiply that by two.

Shelagh: Yes.

Alison: Then I have to multiply the whole of this side by two. So that's everything.

Shelagh: Indeed. Very good.

Alison: Sorry, I'm just going to get some more paper [laughter].

Shelagh: Yeah, I think you need another sheet.

Alison: Oh, it goes on. Um, so ...

Shelagh: Now, we're nearly there.

Alison: I can cancel these two because I'm dividing by two and multiplying by two.

Shelagh: Yes.

Alison: So, I've got m u squared is equal to, and I have to multiply each term in the bracket? ...

Shelagh: Indeed.

Alison: ... by two. So that's the same as m v squared over two multiplied by two. So that just gives me m v squared ...

Shelagh: Yes.

Alison: ... minus two times delta E k.

Shelagh: Great. Yes.

Alison: But ...

Shelagh: You've still got an m.

Alison: I've still got an m. Um, so I just need to divide both sides by m.

Shelagh: Yes.

Alison: And that's the whole of that side by m.

Shelagh: Yes.

Alison: And those ms cancel and I've still got a u squared.

Shelagh: [Laughter]. That's alright. We're homing in on it.

Alison: Aaargh.

Shelagh: Okay, so you've got to get rid of the squared.

Alison: Yeah.

Shelagh: And what is the opposite of squaring?

Alison: Taking the square root.

Shelagh: Yes.

Alison: So, if I take the square root of u squared I'm just left with u and that is equal to, er, m v squared minus two delta E k over m and that's ...

Shelagh: You have to take the square root of the whole thing ...

Alison: ... the whole thing.

Shelagh: You've done it.

Alison: [Laughter]

Shelagh: Well done. And actually, particularly well done because that is more difficult than anything you'll be asked to rearrange in the course. But just by going at it step by step it wasn't actually that bad, was it?

Alison: No, just got to break it down.

Shelagh: Excellent.

Alison: Thank you. [Big sigh].