Cambridge Additional Mathematics

Applications of integration (Chapter 16) 447

Example 8 Self Tutor

A particle P moves in a straight line with velocity function v(t)=t^2 ¡ 3 t+2ms¡^1.

a How far does P travel in the first 4 seconds of motion?

b Find the displacement of P after 4 seconds.

a v(t)=s^0 (t)=t^2 ¡ 3 t+2

=(t¡1)(t¡2)

) the sign diagram ofvis:

Since the signs change, P reverses direction at t=1and t=2seconds.

Now s(t)=

Z

(t^2 ¡ 3 t+2)dt=

t^3

3

¡

3 t^2

2

+2t+c

Hence s(0) =cs(1) =^13 ¡^32 +2+c=c+^56

s(2) =^83 ¡6+4+c=c+^23 s(4) =^643 ¡24 + 8 +c=c+5^13

Motion diagram:

) total distance travelled=(c+^56 ¡c)+(c+^56 ¡[c+^23 ]) + (c+5^13 ¡[c+^23 ])

=^56 +^56 ¡^23 +5^13 ¡^23

=5^23 m

b Displacement=final position¡original position

=s(4)¡s(0)

=c+5^13 ¡c

=5^13 m

So, the displacement is 513 m to the right.

EXERCISE 16C.2

1 A particle has velocity function v(t)=1¡ 2 t cm s¡^1 as it moves in a straight line. The particle is

initially 2 cm to the right of O.

a Write a formula for the displacement function s(t).

b Find the total distance travelled in the first second of motion.

c Find the displacement of the particle at the end of one second.

2 Particle P is initially at the origin O. It moves with the velocity function v(t)=t^2 ¡t¡ 2 cm s¡^1.

a Write a formula for the displacement function s(t).

b Find the total distance travelled in the first 3 seconds of motion.

c Find the displacement of the particle at the end of three seconds.

3 An object has velocity function v(t) = cos(2t) ms¡^1 .Ifs(¼ 4 )=1m, determine s(¼ 3 ) exactly.

1

0

t

+

2

c c+Wec+Ty c+5_Qe

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_16\447CamAdd_16.cdr Monday, 7 April 2014 4:18:18 PM BRIAN