WATERLOO COLLEGIATE INSTITUTE MATHEMATICS SOLUTIONS

WATERLOO COLLEGIATE INSTITUTE
MATHEMATICS
SPH3UW SAMPLE EXAMINATION
SOLUTIONS
SPH3UW January Examination 2008
QUESTION 1
This and the following three questions concern the same physical situation.
A car moves with velocity v0= 12 m/s on a slick road when the driver applies the brakes. The
wheels slide and it takes the car 4 seconds to stop with a constant deceleration.
Which of these pairs of graphs correctly shows the displacement and the velocity of the car during
the time when it decelerates?
(b)
QUESTION 2
How far does the car go before it stops?
(a) 12 m
(b) 24 m
(c) 48 m
1
 v0  v f  t
2
1 m
m
 12  0   4s 
2 s
s
 24m
d 
QUESTION 3
When the velocity of the car has decreased to half its initial velocity the remaining distance is
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SPH3UW January Examination 2008
.
(c) less than half the total distance required to stop.
v 2f  vi2  2ad
62  122  2  3 d
d  18
Remaining distance is 24m -18m =6m
QUESTION 4
What is the kinetic friction coefficient μk between the car and the road surface?
(a) 0.13
(b) 0.31
(c) 0.62
(d) 1.50
(e) 6.00
Fa  F f
ma  uFn
ma  umg
a  ug
a
g
3

9.8
 0.306
u
QUESTION 5
This and the following question concern the same physical situation.
A car starts from rest and, moving with constant acceleration along a straight line, has a speed of 45
m/s after traveling 210 m.
What is the acceleration of the car?
(a) 1.01 m/s2
(b) 2.68 m/s2
(c) 4.82 m/s2
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SPH3UW January Examination 2008
v 2f  vi2  2ad
2
2
 m  m
 45    0   2a  210m 
 s  s
2
 m
 45 
s
a 
2  210m 
 4.82
m
s2
QUESTION 6
How long did the car take to travel the initial 120 m?
(a) 5.44 s
(b) 6.22 s
(c) 7.06 s
(d) 8.71 s
(e) 11.55 s
1
s  vi t  at 2
2
m
 m 1
120m   0  t   4.82 2  t 2
s 
 s  2
2 120m 
m
4.82 2
s
 7.06 s
t
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SPH3UW January Examination 2008
QUESTION 7
This and the following three questions concern the same physical situation.
A student throws a ball off a cliff. When the ball leaves the student's hand it is traveling with a
speed of 23 m/s, and the vertical component of the balls velocity is 18 m/s. The ball hits the ground
after 4.2 seconds.
What is angle did the student throw the ball?
(a) θ = 24°
(b) θ = 37°
(c) θ = 52°
vx  v sin  
18  23sin  
 18 

 23 
 51.5
  sin 1 
QUESTION 8
How long does it take for the ball to reach its maximum height?
(a) 1.8 s
(b) 2.3 s
(c) 3.1 s
a
t

v f  vi
t
v f  vi
a
0  23sin  51.5 
9.8
 1.8s
QUESTION 9
What height h was the ball thrown from?
(a) 9.1 m
(b) 10.8 m
(c) 13.1 m
It took 3.67s to again reach its initial tossing height.
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SPH3UW January Examination 2008
So it took 4.2-3.67s to fall from a height of h with initial vertical speed of 18
1
d  vt  at 2
2
 18  0.53 
1
2
 9.8 0.53
2
 10.9m
QUESTION 10
What is the horizontal distance the ball travels before hitting the ground?
(a) 60 m
(b) 75 m
(c) 97 m
d  vt
 23cos  51 4.2 
 60.8
QUESTION 11
This and the following three questions concern the same physical situation.
Block A is suspended vertically by an ideal string that passes over an ideal pulley and is then
connected to Block B (1.6 kg) that is resting on a ramp with the coefficient of friction μ and with
the incline of 35° as shown in the figure. Block A is observed to accelerate down at a rate of 1.8
m/s2. The tension in the string is measured to be 20 N.
Calculate the mass of block A.
(a) 1.5 kg
(b) 2.0 kg
(c) 2.5 kg
(d) 3.0 kg
(e) 3.5 kg
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SPH3UW January Examination 2008
 F  ma
Fg  FT  ma
m  9.8   20  m 1.8 
8m  20
m  2.5
QUESTION 12
Calculate the frictional force on box B due to the ramp.
(a) 8.1 N
(b) 9.1 N
(c) 12.1 N
(d) 18.1 N
(e) 23.1 N
 F  ma
FT  FF  Fg  ma
20  ma  Fg  Ff
20  1.6 1.8   1.6  9.8  sin  35   Ff
Ff  8.1
QUESTION 13
Calculate the normal force of the ramp on block B.
(a) 8.3 N
(b) 12.8 N
(c) 15.7 N
FN  mg cos  35 
 1.6  9.8 cos  35 
 12.8
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SPH3UW January Examination 2008
QUESTION 14
This and the following question concern the same physical situation.
A ball with mass M = 5 kg is suspended from the ceiling by a wire with tension T1. A student
displaces the mass to the right by pulling, in a horizontal direction, on a second wire with tension T2
= 20 N.
What is the tension T1?
(a) 39 N
(b) 42 N
(c) 48 N
(d) 53 N
(e) 78 N
F
y
0
T1 cos    20  0
F
y
0
T1 sin     9.8  5 
T1 cos    20 T1 sin    49
tan   
49
20
  67.8
T1  52.9
QUESTION 15
What is the angle α?
(a) 37.6°
(b) 42.3°
(c) 55.8°
(d) 67.8°
(e) 84.2°
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SPH3UW January Examination 2008
QUESTION 16
This and the following question concern the same physical situation.
A box of mass M slides down a frictionless inclined plane that makes an angle α with the vertical.
What is the magnitude of the normal force acting on the box?
(a) Mg
(b) Mg sin(α)
careful
(c) Mg cos(α)
QUESTION 17
What is the magnitude of the net force acting on the box?
(a) Mg cos(α)
(b) Mg sin(α)
(c) Mg
QUESTION 18
The cable of a crane is raising a box of mass M = 250 kg with
an upward acceleration of 4 m/s2. What is the tension T in the cable?
(a) 863 N
(b) 1725 N
(c) 3450 N
(d) 6900 N
(e) 9980 N
 F  ma
 FG  FT  ma
FT  ma  mg
 ma  g 
 250  4  9.8 
 3450
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SPH3UW January Examination 2008
QUESTION 19
Harvey is pushing a box A with mass 2 kg which, which, in turn, pushes on another box B with
mass 3 kg. The friction between the blocks and the ground is negligible. If Harvey pushes with a
force F = 15 N, what is the force on block B?
(a) 2 N
(b) 3 N
(c) 5 N
(d) 9 N
(e) 15 N
F2,1 
m2
F
m1  m2
3
15
23
 9N

QUESTION 20
When the tip of the minute hand on a clock is moving past the 12:00 o’clock position, the vector
that gives the direction of the acceleration of the tip is
(a) vector 12
(b) vector 6
(c) vector 3
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SPH3UW January Examination 2008
QUESTION 21
This and the following question concern the same physical situation.
A 3.5kg steel bar in a structural engineering law swings on the end of a rigid steel rod at a constant
speed in a vertical circle of radius 1.2 m, at a frequency of 1.0 Hz
Calculate the magnitude of the tension in the rod due to the mass at the top (A)
(a) 1.3 102 N
(b) 2.0 102 N
(c) 2.5 102 N
(d) 3.0 102 N
(e) 3.3 102 N
 F  ma
c
FT  FG  m4 2 rf 2
FT  m4 2 rf 2  FG
  3.5  4   2 1.2 1   3.5  9.8 
 165.8  34.3
 131.5
QUESTION 22
Calculate the magnitude of the tension in the rod due to the mass at the bottom (B)
(a) 1.3 102 N
(b) 2.0 102 N
(c) 2.5 102 N
(d) 3.0 102 N
(e) 3.3 102 N
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SPH3UW January Examination 2008
 F  ma
c
FT  FG  m4 2 rf 2
FT  m4 2 rf 2  FG
  3.5  4   2 1.2 1   3.5  9.8 
 165.8  34.4
 200.1
QUESTION 23
A car of mass 2.2 103 kg travels around a frictionless banked curve of radius 85m. The banking is
at an angle of 15 to the horizontal. What constant speed must the car maintain to safely travel
around the curve?
(a) 5 m/s
(b) 9 m/s
(c) 15 m/s
(d) 17 m/s
(e) 20 m/s
FN sin    mac
mg tan   
mv 2
r
v  gr tan  

 9.885 tan 15
 14.9
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SPH3UW January Examination 2008
QUESTION 24
This and the following question concern the same physical situation.
The Hubble Space Telescope (HST) follows an essentially circular orbit, at an altitude of 700 km
above the surface of the Earth (radius of Earth is 6.38 106 m ).
Determine the speed (in km/h) needed by the HST to maintain its orbit.
km
h
km
9.45 103
h
km
3.51104
h
km
7.56 103
h
km
2.70 104
h
(a) 1.31104
(b)
(c)
(d)
(e)
v
Gm
r
 6.67 10  5.98 10 
11

24
7.08 106
m
 7.51103
s
km
 2.70 104
h
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