An object is moving along a straiuc69 qdin,j c7ght line such that its velocity, v, $\mathrm{~ms}^{-1}$ , is given by $v(t)=5 t e^{-1.2 t}$ , for $t \geq 0 $.(two decimal places)

1. On the grid below, sketch the graph of v , for $0 \leq t \leq 3$ .

2. Find the distance travelled by the object in the first 3 seconds.
d=  m
3. Find the velocity of the object when its acceleration is zero.
v(0.833)=  ms$^{-1}$

  A particle is moving along a )t duin 282f-u nznql ysj-(8gstraight line. Its velocity can be modelled by the functio-j8gn- ul28yd (t) infun z2qsn v(t)=$2 t-0.3 t^{3}+2$ , for $t \geq 0 $, where v is in $\mathrm{ms}^{-1}$ and t in seconds.
1. Find the acceleration of the particle after 2.2 seconds.
a(2.2)=-   ms${-2}$.
2. 1. Find the time when the particle's acceleration is zero.
t=  s
2 . Find the particle's velocity when its acceleration is zero.
v(1.49)=   ms${-1}$.

  A particle P starts from a fixed point O and m *fcnc/.p4h( rvan*mcoves along a horizontal straight line. Its velocity fh* nnrv4c(a* . m/ccp v $\mathrm{~ms}^{-1}$ after t seconds is given by



The following diagram shows the graph of v .

1. Find the initial velocity of P .
v(0)=  ms$^{-1}$
2. Find the acceleration of P during the first second.
a=  ms$^{-2}$.
3. Write down the number of times the particle changes direction in the first 8 seconds. Justify your answer.
4. Find the total distance travelled by the particle in the first 8 seconds.
d=  m .

  A drone moves in a straight line and its velocity, a9- m6y2 q1b a.dimbnb $v \mathrm{~ms}^{-1}$ , at time t seconds, is given by v(t)=$\left(t^{2}-2\right)^{2}$ , for $0 \leq t \leq 2 $.
1. Find the initial velocity of the drone.
v(0)=  ms$^{-1}$
2. Find the value of t for which the drone is at rest.
3. Find the total distance travelled by the drone in the first 2 seconds.
d=  m
4. Show that the acceleration of the drone is given by a(t)=$4 t^{3}-8 t $.
a(t)=at$^3$-bt; a=  ,b=  .
5. Find the values of t for which the velocity is positive and the acceleration is negative.

  A freight lift moves along a vertical straight line so that its velocity, v \maey pc,a2 ;b-g/du.fp.g*jstl t -s2l ub.je;pdfgag p/ct.y, *hrm{~ms}^{-1} , after t seconds is given by v(t)=$1.5^{t}-4.9$ , for $0 \leq t \leq 6$ .
1. Find when the lift is at rest.
t=  s.
2. Find the acceleration of the lift when t=3 .
a(3)=  ms${-2}$
3. Find the total distance travelled by the lift.
d=   m.

  A particle P moves along a straight line so that its veloci t1nfu*bx;x-ix v uq65ty, $v \mathrm{~ms}^{-1}$ , after t seconds, is given by $v=\sin 3 t-2 \cos t-2$ , for $0 \leq t \leq 6$ . The initial displacement of P from a fixed point O is 5 metres.
1. Find the displacement of P from O after 6 seconds.
s=-  m ; (two decimal places)
The following sketch shows the graph of v .

2. Find when the particle is first at rest.
t=  s; (two decimal places)
3. Write down the number of times the particle changes direction.
hence  ----待选择----12  time(s)
4. Find the acceleration of P after 2 seconds.
a(2)=   ms$^{-2}$.(two decimal places)
5. Find the maximum speed of P .

  Note: In this question, distance is in metres and time is in seconds. An expeb4ik m2rwrqpmz*u1x), 8tbhy8ic 1 ) priment in a particle accelerator moves a particle P in a si*tb ymzphrq 8k4821x m,wirbc)1p) utraight line for six seconds. Its acceleration during this period is given by a(t)=-$2 t^{2}+13 t-15 $, for $0 \leq t \leq 6$ .
1. Write down the values of t when the particle's acceleration is zero.
t=  and 5.
2. Hence or otherwise, find all possible values of t for which the velocity of P is increasing.

The particle has an initial velocity of $7 \mathrm{~ms}^{-1}$ .
3. Find an expression for the velocity of P at time t .

4. Find the total distance travelled by P when its velocity is decreasing.
d=  m

  A remote-controlled car C is moving along a straight lp l 21d0 /eyp)jqc-/laez+d cpath so that its velocity, qdjep l p+l 0deca1/lc/y-2 )z v, $\mathrm{~ms}^{-1}$ after t seconds, is given by v=$2 \sin t-\cos 5 t+0.1$ , for $0 \leq t \leq 4 $. The initial displacement of C from a fixed point O is 2 metres.

1. Find the displacement of C from O after 4 seconds.
s=  m.
2. Find the second time for t , when the particle is at rest.
t=  s.
3. Write down the number of times C changes direction.
4. Write down the number of times C is neither accelerating or decelerating.
5. Find the maximum distance of C from O during the time 0 \leq t \leq 4 and justify your answer.
d$_{max}$=  m

  John drops a stone from the top of a cliff which is h metres abj06 dhrat 5wi(ove sea level. The stone s r6(j5 dwi0htatrikes the water surface after 9 seconds.
The velocity of the falling stone, $v \mathrm{~m} \mathrm{~s}^{-1}$, t seconds after John releases it, can be modelled by the function(round number)



1. Find the velocity of the stone when t=12 , giving your answer to the nearest $\mathrm{m} \mathrm{s}^{-1}$ .
v(12)≈   ms$^{-1}$.(round number)
2. Calculate the value of h , giving your answer to the nearest metre.
h≈   m,(round number).
The velocity of the stone when it reaches the bottom of sea is $10 \mathrm{~m} \mathrm{~s}^{-1}$ .
3. Determine the depth of sea near the cliff, giving your answer to the nearest metre.
d≈   m.

  An object moves in a straight line such that itfb6o8qgav,j s 6oi -,ys velocity, v \mathrm{~m} \mathrm{~s}^{-1} , at time t seconds, is given by


1. Find the value of t , for t>0 , when the object is instantaneously at rest.
t=  s.
The object returns to its initial position at t=T .
2. Find the value of T . Give your answer correct to three significant figures.
T≈  s.(one decimal place)

  A particle moves along the x -axi4) bzvjadxu 6ebm59+x l*u4kqs so that its velocity, $v \mathrm{~m} \mathrm{~s}^{-1}$ at time t seconds, is given by the equation

$v(t)=10 e^{-\frac{t}{4}}-5$

1. Find the time at which the particle changes direction.
The particle changes direction after approximately (  .77) seconds.
2. Find the magnitude of the particle's acceleration at time t=2 seconds.

The particle starts from the origin O .
3. Find an expression for the displacement of the particle from $\mathrm{O}$ at time t seconds.
$s(t)=a-b e^{-\frac{t}{4}}-c t$ ; a=  ,b=  ,c=  .
4. Find the time at which the particle returns to the origin.
5. Calculate the total distance travelled by the particle by the time it has returned to the origin.
the time the particle has returned to the origin, it has travelled approximately (  .3) metres.