An object is moving /z 6wy(up oh8balong 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 lij,a h5c9nb.myne. Its velocity can be modelled by the fu.yjm,nhb 5ac 9nction 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 fi hkqa9a(xe* hq2,7eyz u8id x3xed point O and moves along a horizontal straight li87 qx2ax hue9y3h(dz ke,iqa*ne. Its 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 straightcgoo:+ e/j9kn 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 a.5 *ilaj gk wa5eiq51rlong a vertical straight line so that its velocity, v \mathrm al5e 5a jk*iq5ir1g.w{~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 moves7lb97(j(jwfy1 -qanqv+ mnqq along a straight line so that its velocity, $v \mathrm{~ms}^{-1}$ , 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 exxfr9ru. upg*d;0hm 2rperiment in a particle accelerator moves a particle P in a straight line for six secx u0 p2r;.r 9r*hfmgduonds. 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 alvgokpq:kav d9 nw4.mj3 )l )k/sq2ia; ong a straight path so that its velocitys;q)2 ngd9l i.kp4oq)a:k vma/jvw3k, 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 above sea leve-ub((ag5z6 cuf h6d mel. The stone strikes huadeg6 c6(f mb(5u-zthe 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 its velocity, v \mat/ 6/mnp8z;9xjsfba l .nh:gemhrm{~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,hugw 6twdzw:crw pc8sd *;2-, $v \mathrm{~m} \mathrm{~s}^{-1}$ 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.