Cambridge Additional Mathematics

300 Vectors (Chapter 11)

Example 15 Self Tutor

An object is initially at (5,10) and moves with velocity vector 3 i¡j metres per minute. Find:

a the position of the object at timetminutes

b the speed of the object

c the position of the object at t=3minutes

d the time when the object is due east of (0,0).

a r=a+tb

)

μ

x

y

¶

=

μ

5

10

¶

+t

μ

3

¡ 1

¶

, t 2 R

)

μ

x

y

¶

=

μ

5+3t

10 ¡t

¶

Aftertminutes, the object is at (5 + 3t, 10 ¡t).

b The speed of the object is jbj=

p

32 +(¡1)^2 =

p

10 metres per minute.

c At t=3minutes, 5+3t=14and 10 ¡t=7. The object is at (14,7).

d When the object is due east of (0,0), ymust be zero.

) 10 ¡t=0

) t=10

The object is due east of (0,0) after 10 minutes.

EXERCISE 11H

1 A particle at P(x(t),y(t)) moves such that x(t)=1+2t and y(t)=2¡ 5 t, t> 0.

The distances are in centimetres andtis in seconds.

a Find the initial position of P.

b Illustrate the initial part of the motion of P where t=0, 1 , 2 , 3.

c Find the velocity vector of P.

d Find the speed of P.

2aFind the vector equation of a boat initially at (2,3), which travels with velocity vector

μ

4

¡ 5

¶

.

The grid units are kilometres and the time is in hours.

b Locate the boat’s position after 90 minutes.

c How long will it take for the boat to reach the point (5,¡ 0 :75)?

3 A remote controlled toy car is initially at (¡ 3 ,¡2).It

moves with constant velocity 2 i+4j. The distance units

are centimetres, and the time is in seconds.

a Write an expression for the position vector of the car at

any time t> 0.

b Find the position vector of the car at time t=2: 5.

y

x

a

r

A,() 510

R

O

= 3 _____

-1______

_____ _

b {}_

cyan magenta yellow black

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_11\300CamAdd_11.cdr Monday, 6 January 2014 9:58:12 AM BRIAN