The Chemistry Maths Book, Second Edition

118 Chapter 4Differentiation

in which y

1

,y

2

, andX 1 = 1 x

1

1 + 1 x

2

are constant. Then

= 1 0 for a minimum

Hence Snell’s law.

0 Exercises 84 – 86

4.11 Linear and angular motion

The description of the motion of bodies in space is an important application of the

differential calculus. We consider here only the simplest kinds of motion; motion in a

straight line and motion in a circle. More general kinds of motion are discussed in

Chapter 16.

Linear motion

Consider a body moving in a straight line, along the x-direction say. Let O be a fixed

point and let P be the position of the body at time t. The distanceOP 1 = 1 xis then a

function of time;x 1 = 1 f(t).

If the body moves from point xto pointx 1 + 1 ∆xin time interval ∆t, then the

average rate of change of xin the interval is

The limit of this as∆t 1 → 10 is the instantaneous rate of change of xwith respect to t. It

is the linear velocity, or simply the velocity, at time t,

(4.25)

When vis positive, xis increasing and the body is moving to the right. When vis

negative, xis decreasing and the body is moving to the left. Velocity is in fact a vector

quantity, having both magnitude and direction; vectors are discussed in Chapter 16.

The magnitude of the velocity is the speed. The derivative of the velocity is the

acceleration,

acceleration== (4.26)

d

dt

dx

dt

v

2

2

velocity==v

dx

dt

∆

∆

∆

x

t

=average velocity in interval t

dt

dx

dr

dx

dr

dx

x

r

x

r

11

1

12

2

11

1

12

2

2

1111

=+ =−=

vv vv

siinθθsin

1

1

2

2

vv

−

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

• •

o x p

x

.....

........

.........

.........

.......

Figure 4.12