The Manga Guide to Physics

206 Chapter 4 Energy

Of course, this is true for every segment i, so we can add up all of these to find the

work done across the entire displacement, from x

1

to x:

( ( (

1

2

2

2

1

2

1

2

1

2

3

2

1

2

2

2

1

2

4

2

1

2

3

2

1

mv mv mv mv mv mv

F x F

- - -) + ) + ) + ... =

+∆

22 3

∆ ∆x F x+ + ...

By looking closely at the left-hand side, you can see that most of the terms will cancel!

We are left with just two terms:

1

2

2

1

2

1

2

mv mv

n

-

So we can rewrite this equation as:

1

2

2

1

2

1

2

1

mv mv F x

n i

i

n

- =

=

∑

∆

We have added up the little pieces of work done at each instant to get the total change

in energy—the work W. You’ll see that this looks awfully similar to the definition of an inte-

gral. It ...