13 May M1 International Model Red

8/11/2019 13 May M1 International Model Red

1/8

Ht

MftY

2ot3

rNEeNfrrtoNr\L

1.

Two

particles

I and

B,

of

mass 2 kg and 3 kg respectively, are moving towards each other

in

opposite directions along

the same straight line

on

a smooth horizontal surface. The

parlicles

collide

directly.

Immediately before the

collision

the speed ofl is 5 m s

I

and

the

speedofB

is 6 m

s-r.

The magnitude

of the impulse

exertedonBbyl

is

14Ns.

Find

(a)

the

speed of ,4 immediately

after the

collision,

(3)

(b)

the

speed

of

B immediately after the collision.

(3)

ft-s

a

Vp

lYlorn

A

blare-'

to

e

lmou[se-

--

r+

ln.^$

ger

-

Lxln

ci-Il

SxL+

Ln-6

=

Lr-Z+

3V6

-bB

-

o_

Vc

2Vn

=

-tf

vft.4

-

=>

-8 =-9+3Vg

=)

3Vs=

-*

...

Vg't


2/8

2-

Figure

I

A

particle

of

weight 8

N is attached

at C to

the

ends

of

two light

inextensible strings

lC

and.BC.

The other

ends,

I

and

-8,

are attached

to a fixed

horizontal ceiling.

The

particle

hangs

at rest in equilibrium,

with

the strings

in

a vertical

plane.

The string

lC

is

inclined

at

35'

to the

horizontal and

the string

BC is

inclined

at25"

to

the

horizontal, as shown

in

Figure 1. Find

(i)

the tension

in

the

string

lC,

(ii)

the tension

in

the string

BC.

(8)

(oStn'LS

Ts

(os2S

=)

TB= TA-C"sS

Cos2,S

*8

fr+4--O


3/8

3.

6

-{

t^"

5rP-'$3.148196

'>

{nnar

=7r,tNA

- 11'6

k

+"

.$:"*

ag

Figure

2

A fixed rough

plane

is

inclined

at 30o to the

horizontal. A

small

smooth

pulley

P

is

fixed

at the top of

the

p1ane.

Two

parlicles

A

and

B,

of mass

2 kg and 4 kg respectively, are

attached

to the ends of a

light

inextensible string

which

passes

over the

pulley

P.

The

part

of

the string

from I to P is

parallel to a line of

greatest

slope of

the

plane

and

B hangs

freely below P, as shown

in Figure 2.

The coefficient of

friction between

I and the

plane

i,

*.

tritiutty

z is held

at rest on the

plane.

The

parlicles

are

released

from

rest with

3

the string taut and

I

moves up

the

plane.

Find the tension

in

the

string

immediately after the

particles

are

released.

(e)

\,Ld4_

s.Xstcx-

frh:

t'6^

=)

{"{o-{

Sre

}ur^-.{Lo-

pL"r-

=

30.

45-T

--

k".

o

+g-r-

f3

T=

*aru

r

-b):

=

-:

^,

*b


4/8

4. At time r

:

0,

two

balls

I

and

B are

projected vertically upwards. The ball

I

is

projected

vertically upwards

with speed

2 m s

1

from

a

point

50

m above

the

horizontal

ground.

The ball

-B

is

projected

vertically

upwards

from

the

ground

with speed

20 m s

1.

At time

l: 7 seconds,

the

two

balls

are

at the same

vertical height, h metres, above

the

ground.

The

balls

are modelled

as

particles

moving freely

under

gravity.

Find

(a)

the

value

of

7,

(s)

(b)

the

value

of

ft.

(2)

S

.

-

(so-h)

Lt,

-

2^

V

(

=

-1.8

L,

r

---q.8

:{

s

t{

V

A

t

.I

,20

g

=

6t+ atl -4

h-

50

=

LT

-

*-\ro

h

--

eor-

+'1T"

=)

/.

SO

=x

+/-rcf

:)

-SO=

-lgr

o

@

*.1T1,

."

T=7

b)

h=

2o(?)-

q,r(?)'

,

Eb


5/8

5.

Figure

3

A

particle P of

mass 0.6 kg slides

with

constant

acceleration

down a line

of

greatest

slope

of

a

rough

plane, which

is

inclined at 25"

to the

horizontal.

The

particle passes through

two

points

A

and

B, where

AB: 10

m,

as shown in Figure 3.

The

speed

of

P at A is

2 m

sr.

The

particle

P takes

3.5

s

to

move

from

I

to

B.

Find

(a)

the

speed

of

p

at

B, (3)

(b)

the acceleration

of P,

e)

(c)

the coefficient

of friction

between P

and

the

plane.

(5)

{r***__ltNr

NK--

s.3z1o6qtr'

ts\

^

f

"o'b59a2s

-'

{L^r

=

s'j21oE981^

0.53tosZs

.f\

=r,rc.

z)

L'+8+715

-S'3210818110

S

--

lO

S=tr"t

+ aL'--)

tO

'++fo.(s.s)'

(r

2-

v

b)

..-^,y

avt

t

=3.S

a)

V

=tr.+at

V=2-+(H,X|)

.

+

c)

a.+8tr11s

-

o

6(T)=

s'321o81trl,r

.'-

),\--

O.+l

(zsp)

blanl

c\.


6/8

6,

lln

this

question

i

and

j

are horizontal

unit

vectors due east and

due north

respectively.

Position

vectors ore

given with

respect

to a

fixed

origin O.)

A ship 5

is moving

with constant

velocity

(3i

+

3j)

km h

r.

At time t: 0,

the

position

vector

of

S is

(-4i

+

2j) km.

(a)

Find the

position

vector

of

S

at time

I hours.

(2)

A ship 7

is

moving

with

constant

velocity

()i

+

ni)

km h

r.

At

time

I

:

0, the

position

vector of

7 is

(6i

+

j)

km. The two

ships

meet

at the

point

P.

(b)

Find the

value

ofr.

(s)

(c)

Find

the

distance

OP.

(4)

67

=64"

")

v

'

(e)

s=

(-I).

b)

r=(t)--(l)

fi

2+3t

ol+tn

-)

2-+L=

c)

P"

(-;)-

(t

)

=

(;

)

.

(i)

2b

=

-S+3t"

=

5b

t'L

-,

+2n

i

2a:

$ =)fPlS

=

8.2Sha,.

(3sg)

----


7/8

1

Figure

4

A

truck of

mass 1750

kg

is

towing

a

car

of

mass

750

kg

along

a

straight

horizontal

road.

The

two

vehicles

are

joined

by

a light towbar

which is inclined

at

an angle d to the

road,

as shown in

Figure 4.

The vehicles

are

travelling at 20

m

s

I

as they

enter a

zone where

the speed

limit

is

14 m s

1.

The

truck's

brakes

are applied to

give a constant braking

force

on the

truck.

The distance travelled

between

the instant when

the brakes are

applied and

the instant when

the

speed

ofeach

vehicle

is

14

m s

I

is 100

m.

(a)

Find the

deceleration

of the truck

and

the car.

(3)

I

u

v

a\

t

+)

A.

-l'OL

?''-

Note

-

6lls*

bre

tL'ng

tprr,rtorr-

rs'TH(I)ST

r-oz

4-

r. oL

()

urt\dLz-s")s+e$r

R+

S@

+360

+o.qT

-O-tr

.2S00y1.02

=1

(tt6O

2

2SSo

:.

[2,

i45ON

.-...._-.....-

=)

T-hcust.

5l+.,

(ssp)

- -.--.-.-.

1

16()

V

2

,

(,tZ+,l*a.S

t1[

-

*OD+ZOOa

=)

?-@a= -2OY

The

constant

braking

force on

the truck

car also

experience

constant

resistances

Given

that cos d

:

0.9,

find

(b)

the

lorce

in

the towbar,

(c)

the value

ofR.

=20

=

tg

has magnitude

R newtons.

The

truck

and the

to motion of 500

N and

300

N respectively.

(4)

(4)

b)

Car

3oO

+

o.1T

=?E>Q-oz)


8/8

R

tr

D

.

t'

C

A

0.2m

2m

Figure

5

A uniform rod

lB

has length 2 m and mass

50 kg. The rod is in equilibrium

in

a

horizontal

position,

resting

on two smooth supports

at C

and D,

where AC

:

O.2

metres

and

DB: x

metres,

as

shown

in Figure 5. Given

that

the

magnitude

of the

reaction on

the

rod

at D

is

twice

the magnitude

of the reaction

on the rod at C,

(a)

find

the

value

ofx.

(6)

The

support

at

D is

now moved

to

the

point

E on the

rod,

where

EB

:

0.4

metres. A

particle

of

mass

m

kg is

placed

on the

rod at B, and

the

rod

remains

in equilibrium

in a

horizontal

position.

Given

that

the

magnitude

of the reaction on the

rod at E

is

four

times

the magnitude

ofthe reaction on

the

rod

at

C,

(b)

find

the value

of m.

A

,

so3

.xm

a)

t=

il.

,1,

+

3r= sga

L.

9q

3u

-) *{*

=

Lo4

so9

Bt

4r(yp.k+(x1.8

"

SOSxl

C*z

a

$5x

I's

=

5o3r

t

r.)

t2SO.

ol=

32OJ

+

r.^{

\aul1p,

b)

.'.

k--

Ea

l+J

z(=

rya.

2q

s

Q.6

f"\

16)

'-)

3'$e--

SoS

t.-V

-)

$r(

=

SO5'i

n^5

Documents

13 May M1 International Model Red