CONTOH 10
A particle starts from a fixed point O, and moves along a straight line. After t seconds, its displacement,s is given by s=t3-3t2-4t. Calculate
Suatu zarah bermula dari titik tetap O dan bergerak di sepanjang satu garis lurus. Selepas t saat, sesarannya, s meter diberi oleh s=t3-3t2-4t. Hitung
a) the initial acceleration, in ms-2, of the particle,
pecutan awal, dalam ms-2, zarah itu,
b) instantaneous acceleration of the particle, in ms-2 , after 5 seconds,
pecutan seketika zarah itu, dalam ms-2, pada masa 5 saat,
c) the acceleration of the particle,in ms-2, when it passes through the fixed point O again,
pecutan zarah itu, dalam ms-2, apabila melalui titik O semula,
d) the range of values of t, in seconds,when the acceleration of the particle is positive
julat nilai t, dalam saat, apabila pecutan zarah itu ialah positif
Solution
Penyelesaian
Given the displacement function, s=t3-3t2-4t.
Then, the velocity function v==3t2-6t- 4 and acceleration function, a==6t-6.
Diberi fungsi sesaran, s=t3-3t2-4t.
Jadi, fungsi halaju, v ==3t2-6t- 4 dan fungsi pecutan, a = =6t-6
a) When t=0,a=6(0)-6=-6ms-2
Hence, the initial acceleration is -6ms-2 .
Apabila t=0,a=6(0)-6=-6ms-2
Maka, pecutan awal zarah ialah –6 ms-2.
b) When t=5,a=6(5)-6=24ms-2
Hence, the instantaneous acceleration at 5 seconds is 24 ms-2 .
Apabila t=5,a=6(5)-6=24ms-2
Maka, pecutan seketika zarah itu pada masa 5 saat ialah 24 ms-2.
c) When the particle passes through the fixed point O again,
s=0
t3-3t2-4t=0
t(t2-3t-4)=0
t(t-4)(t+1)=0
t=0,t=4,t=-1(rejected)
When t=4,a=6(4)-6=18ms-2
Hence, when the particle passes through fixed point O again, the acceleration is 18 ms-2 when t=4s.
Apabila zarah melalui titik O semula,
s=0
t3-3t2-4t=0
t(t2-3t-4)=0
t(t-4)(t+1)=0
t=0,t=4,t=-1(ditolak)
Apabila t=4,a=6(4)-6=18ms-2
Maka, apabila zarah itu melalui titik O semula, pecutannya ialah 18 ms-2 semasa t=4s.
d)For the acceleration to be positive,
a>0
6t-6>0
6t>6
t>1
Hence, the acceleration would be positive when t>1.
Pecutan zarah positif,
a>0
6t-6>0
6t>6
t>1
Maka, pecutan zarah adalah positif untuk t >1.