The Chemistry Maths Book, Second Edition

152 Chapter 5Integration

5.7 Dynamics

Velocity and distance

As in Section 5.1, we consider a body moving along a curve from point Aat timet 1 = 1 t

A

to point Bat timet 1 = 1 t

B

, and let the distance from Aalong the curve bes(t)at some

intermediate time t. By the discussion of Section 4.11, the quantityv(t) 1 = 1 ds 2 dtis the

velocity of the body at time t. The differential lengthds 1 = 1 vdtis the distance travelled

in the infinitesimal time intervaldt, and the total distance is then

(5.48)

EXAMPLE 5.15A body falling under the influence of gravity

A body of mass mfalling freely under the influence of gravity experiences the constant

accelerationdv 2 dt 1 = 1 g(air resistance and other frictional forces are neglected). Then

v(t) 1 = 1 gt 1 + 1 v(0), where v(0)is the velocity at timet 1 = 10. If the body falls from rest at

t 1 = 10 , thenv(t) 1 = 1 gt. The distance travelled in the time intervalt 1 = 1 t

A

tot 1 = 1 t

B

is then

Therefore, given thatg 1 ≈ 1 9.8 m s

− 2

, the distance travelled in the first second of fall is

4.9 m, and in the following second it is4.9 1 × 1 (2

2

1 − 11

2

) m 1 = 1 14.7 m.

0 Exercise 52

Force and work

Consider a body moving along the xdirection between points Aand Bwith velocity

v 1 = 1 dx 2 dt. If a force Facts on the body then, by Newton’s second law, the acceleration

aexperienced by the body is given by

(5.49)

where mis the mass of the body. Work is done on the body by the application of

the force. If the force is constant then the work done is (work 1 = 1 force 1 × 1 distance)

W 1 = 1 F(x

B

1 − 1 x

A

).

If the force is notconstant between Aand B, but is a function of position, F 1 = 1 F(x),

then the work done is obtained by means of the integral calculus. The work done

on the body between positions xandx 1 + 1 ∆xis∆W 1 ≈ 1 F(x) ∆x. In the limit∆x 1 → 10 , the

corresponding element of work isdW 1 = 1 F(x) 1 dxand the total work done is the integral

WdW Fxdx (5.50)

W

x

x

==ZZ

0

A

B

()

Fmam

d

dt

==

v

s t dt gt dt g t t

t

t

t

t

===−











ZZ 

A

B

A

B

BA

v()

1

2

22

sds tdt

s

t

t

==ZZ

0

A

B

v()