Higher Engineering Mathematics, Sixth Edition

258 Higher Engineering Mathematics

Thus,the resultant of the two accelerations is a single

vector of 31.76m/s^2 at 118. 18 ◦to the horizontal.

Problem 10. Velocities of 10m/s, 20m/s and

15m/s act as shown in Fig. 24.26. Calculate the

magnitude of the resultant velocity and its direction

relative to the horizontal.

20 m/s

10 m/s

15 m/s

158

308

(^1)
(^2)
(^3)
Figure 24.26
The horizontal component of the 10m/s velocity=
10cos30◦= 8 .660m/s,
the horizontal component of the 20m/s velocity is
20cos90◦=0m/s,
and the horizontal component of the 15m/s velocity is
15cos195◦=− 14 .489m/s.
The total horizontal component of the three velocities,
H= 8. 660 + 0 − 14. 489 =− 5 .829m/s
The vertical component of the 10m/s velocity=
10sin30◦=5m/s,
the vertical component of the 20m/s velocity is
20sin90◦=20m/s,
and the vertical component of the 15m/s velocity is
15sin195◦=− 3 .882m/s.
The total vertical component of the three forces,
V= 5 + 20 − 3. 882 =21.118m/s
From Fig. 24.27, magnitude of resultant vector,
R=
√
H^2 +V^2 =
√
5. 8292 + 21. 1182 =21.91m/s
The direction of the resultant vector,
α=tan−^1
(
V
H
)
=tan−^1
(
21. 118
5. 829
)
= 74. 57 ◦
5.829
21.118
R
 
Figure 24.27
Measuring from the horizontal,
θ= 180 ◦− 74. 57 ◦= 105. 43 ◦
Thus,the resultant of the three velocities is a single
vector of 21.91m/s at 105. 43 ◦to the horizontal.
Usingcomplex numbers, from Fig. 24.26,
resultant= 10 ∠ 30 ◦+ 20 ∠ 90 ◦+ 15 ∠ 195 ◦
=(10cos30◦+j10sin30◦)
+(20cos90◦+j20sin90◦)
+(15cos195◦+j15sin195◦)
=( 8. 660 +j 5. 000 )+( 0 +j 20. 000 )
+(− 14. 489 −j 3. 882 )
=(− 5. 829 +j 21. 118 )Nor
21.91∠ 105. 43 ◦N
as obtained above using horizontal and vertical
components.
The method used to add vectors by calculation will
not be specified – the choice is yours, but probably
the quickest and easiest method is by using complex
numbers.
Now try the following exercise
Exercise 103 Further problems on
addition of vectors by calculation

1. A force of 7N is inclined at an angle of 50◦
   to a second force of 12N, both forces act-
   ing at a point. Calculate magnitude of the