An object is moving m9(hqbd): coxh:aoqex 46(zz along a straight 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 straight line. Its velocity can be modtl s(nqgfpz6t(54d2 tzb2 xq,elled by the function (dn(tq4q 26szb5lftt g2,xpz 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 andsccz gv /7hv *:wivi,/d0 k8yp moves along a horizontal straight line. Ich7ki/,:z viy* wg8sd 0vc/vpts velocity 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 straigiec828b-e-a8 dfz h.id q0,r obkiym0ht line and its velocity, $v \mathrm{~ms}^{-1}$ , 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 gj*uu/e (vs:kvelocity, v \mathrmugke us:j/v(* {~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 vel(8iah:q;dm j,hqdi ;pocity, $v \mathrm{~ms}^{-1}$ , 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 expe1 ojmdw di))1w(x+r5qt ps 58ftbmdt *riment in a particle accelerator moves a particle ptm 5fd+ q*tr8 osx)1di )jwb15tm(wdP in a straight 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 ugh35 2 l7avlk is moving along a straight path so that its velocity, v,v7la hgk325ul $\mathrm{~ms}^{-1}$ 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 ciae7c 0jqm(t;( xfq /zliff which is h metres above sea level. The stone strikes the water surface after efiat z qj/c;((qm x709 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 its velocity,1g;, *rskwje6nck id0 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 -axis so that its velocity,(j v*q gmcyvyn0g0r 4a6 qfoj(2u4p. s $v \mathrm{~m} \mathrm{~s}^{-1}$ 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.