3 . A particle moves in a straight line . Its initial displacement is one metre . Its velocity , v ms
� 1
, at time t seconds , is given by
11 v( t) � �cos�t , for 0�t
� 5.
( a ) ( i ) Use a suitable substitution , show that s( t) � � cos udu .
( ii ) Hence , find the expression of st (), the displacement of the particle .
( b ) Find the values of t for which the displacement of the particle is 0 m .
( c ) ( i ) Find the expression of at (), the acceleration of the particle .
[ 6 ]
[ 3 ]
( ii ) Hence , find all possible values of t for which its acceleration is positive . [ 5 ]
It is given that there are five stationary points of the graph of st (), for 0�t
� 5.
( d ) Write down the number of times for which the particle is at rest . [ 1 ]
4 . A particle moves in a straight line . Its initial displacement is zero metres and its initial velocity is
1
48 ms � . Its acceleration ,
2 a ms � , at time t seconds , is given by
3 2 a( t) � 4t �33t �88t
� 76 , for 0�t
� 5.
( a ) Find the expression of vt (), the velocity of the particle .
( b ) Hence , find the expression of st (), the displacement of the particle .
( c ) Find all possible values of t for which the displacement is positive and the acceleration is negative .
The table below shows the information of the graph of vt (), for 0�t
� 5.
[ 4 ]
[ 4 ]
[ 4 ]
t |
0�t �
2
|
t �
2
|
2�t �
3
|
t �
3
|
3�t �
4
|
t �
4
|
4�t �
5
|
|
vt
()
|
positive |
0 |
positive |
0 |
negative |
0 |
positive |
( d ) ( i ) Write down the number of times for which the particle is at rest .
( ii ) Write down the number of times for which the particle changes its direction .
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