Unit 1 Lesson 1
Exercises
1. A train 300. metres long is travelling at 25.0 m/s when it passes a 100. m long
train travelling at 10.0 m/s in the same direction. How long does it take for the
longer train to completely pass the shorter?
(26.7 seconds)
2. How long does it take for these two train to completely pass each other?
10.0 m/s
120. m long
6.00 m/s
80.0 m long
260. m
(28.8 sec)
3. Rocket cars drive over a 1.0 kilometer long course and try to break the
sound barrier (330 m/s). How long would it take such a car to travel the length
of the track?
(3.0 seconds)
4. The Earth travels around the sun in a circle of radius 150. million km and
does a complete cycle in about 365 days. Find its speed.
(29 800 m/s)
→
5. Fill in the chart and graph for v = (+5 î ) m/s
(-8 î ) metres ( 8m West)
time
position
0 seconds
(-8 î ) metres
1 second
with a starting position of
2 second
3 second
t=0
x(metre)
position
10
0
10
20
x(metre)
position
t
6. What does it mean to say that an object has
a) negative position?
b) negative velocity?
7. Give an example (draw a diagram) of an object with positive position and
negative constant velocity
Physics 12 Student Workbook
©Donald J. Mathewson 2002
9
8. Is the magnitude of speed always the same as the magnitude of velocity?
YES
or
NO
Explain your answer using relevant principles of physics.
___________________________________________________________________
___________________________________________________________________
___________________________________________________________________
9. Mix and Match speeds
human walking
fastest human running
top fighter jet
light
car on city street
formula-1 race car
sound
fastest animal
terminal velocity-human
330 m/s
220 kph
4 kph
10 m/s
50 kph
3600 kph
300 000 km/s
55 m/s
100 kph
10. Consider the following velocity-time graph. Find the indicated area
together with units and explain what this area tells you. (12.5 m distance)
v (m/s)
velocity
2.5 m/s
area
1s
2s
3s
4s
5s
t
11. How long does it take light (300. million metres per second) to travel from
the Sun to the Earth (150. million km)? Answer in minutes (8.33 minutes)
12. If you glance away for 0.75 seconds while you are travelling on the
highway at 120 kph, how far does the car travel in that time? ( 25 m)
Physics 12 Student Workbook
©Donald J. Mathewson 2002
10
Unit 1 Lesson 2
Exercises
1. A car travels at 20.0 m/s for 12.0 seconds, then 10.0 m/s for 20.0 seconds then
30.0 m/s for 6.00 seconds (all in the same direction)
a) sketch a graph of the car's speed vs time
b) sketch a graph of the car's position vs time
c) find the total distance travelled
(620. m)
d) find the average speed of the car
(16.3 m/s)
2. A car leaves Vancouver for Hope, a total distance of 175 km. The driver needs
to complete the trip in 3.00 hours, but spends the first 100. kilometers in traffic
moving at 40.0 kph. How fast does the driver need to move to complete the trip
on time?
(150. kph)
3. A car travels East for 20.0 seconds at 4.00 m/s, then West at 5.00 m/s for 75.0
metres, then East again for 15.0 seconds, covering 60.0 metres.
a) find the total distance
(215 m)
b) find the total displacement
(65.0m East OR (+65.0 î ) metres
c) find the average speed
(4.30 m/s)
d) find the average velocity
(+1.30 î ) m/s
4. Consider the instruments on your car dashboard
a) does the speedometer really measure speed, or does it measure velocity?
b) explain your answer to a)
c) does the odometer measure distance or displacement?
d) explain your answer to c)
5. Describe the motion depicted by this graph and fill in the missing graph
x
v
t
t
6. An object moves 100. km to the East at 30.0 kph, then moves West for 120.0
minutes. If the average speed for the trip is 35.0 kph, then where does the
object end up? (assume it started at position 0.0 metres) ( 13.3 km East or +î )
7. A car travels 120. km East at 45.0 km/hr, then goes West 50.0 km at 70.0
km/hr, then goes East again for time t at 40.0 km/hr. If the average velocity
for the entire trip is 30.0 km/hr East, find the time t
(3.14 hrs)
8. Sketch a v-t and x-t graph for #7.
9. A truck goes up a long hill at 40 kph and down the other side (equal length)
at 60 kph. Which of the following could be the average speed?
a) 50 kph
b) 48 kph
c) 52 kph
Explain your answer using relevant physics principles
Physics 12 Student Workbook
©Donald J. Mathewson 2002
14
Unit 1 Lesson 3
Exercises
1. A dropped object will start at rest and accelerate due to gravity:
a = (-9.80 ^j) m/s 2
a) Fill in the velocity chart below for a dropped object
b) graph velocity-time
time
velocity
0 second 1 second 2 second 3 second 4 second
v
t
2. Describe the motion and produce the missing graph.
x
v
t
t
c)
x
v
t
t
3. What happens to the motion of an object if
a) the object is moving East and the acceleration is West
b) the object is falling and accelerating upwards
c) the object is moving up and accelerating up
Physics 12 Student Workbook
©Donald J. Mathewson 2002
19
4. Describe the motion represented by this graph
v
2
5
t
5. For the previous graph, determine
a) greatest distance from origin
b) displacement between 1s and 3s
c) distance travelled between 0s and 4s d) velocity at 3 s
e) average velocity between 0 and 4 s. f) maximum velocity
g) acceleration at 3 s
h) avg acceleration between 3s and 5s
(6m: 3.5 m; 6 m; 1 m/s; 1.5 m/s; 2 m/s; -1m/s^2; -1m/s^2)
6. At what times is this object
a) moving East
b) moving West
c) stopped
v
5
t
7. Which of the following best represents the position-time graph for #6
x
a)
b)
c)
x
x
t
x
t
t
d)
t
8. Produce graphs of position-time and velocity time for an object that starts at
rest, speeds up to a maximum speed and then immediately begins slowing down
to rest.
9. A car moving at 20.0 m/s East decelerates (accelerates opposite to the
direction of motion) until the car stops. If the deceleration is 4 m/s2, produce a
velocity time graph showing how long it takes the car to stop.
Physics 12 Student Workbook
©Donald J. Mathewson 2002
20
Unit 1 Lesson 4
Exercises
1. A dolphin accelerates from 1.0 m/s to 7.6 m/s in 5.5 seconds. Find
a) the acceleration b) the distance travelled
(1.2 m/s2; 24 m)
2. A car moving at 30.0 m/s to the right slows down (accelerates to the left) at
9.80 m/s2. How far does the car go before stopping?
(45.9 m)
3. A car travels at 15 m/s and speeds up uniformly. Over the next 5 seconds, it
travels 85 m. Find the acceleration
(0.8 m/s2)
4. A bungee jumper moving down at 5 m/s ends up moving upwards at 8 m/s
2.0 seconds later.
a) find the acceleration
(+6.5 m/s2 ^j)
b) why is the acceleration not 1.5 m/s2 ?
5. A car starts at rest and accelerates in 4 seconds to a speed of 20 m/s, which it
maintains for the next 8 seconds. Find the total distance travelled. (200m)
6. A train starts at rest and accelerates at 0.250 m/s2. After 5.00 seconds, find its
speed and distance travelled.
(1.25 m/s; 3.12 m)
→ → →
7. a) Substitute the velocity formula vf = vi + a∆t
→ →
→ v f +vi
into the displacement formula d =
∆t
2
→ →
1→
and prove that d = vi t + a ∆t2
2
b) interpret this displacement formula
8. A child slides down a playground slide, accelerating from rest and
travelling 6.00 metres in 4.50 seconds. When they hit the ground, their legs
bend 20.0 cm as the child stops. What is the child's acceleration in stopping?
(17.7 m/s2)
9. How long does it take a 150m long train moving at 120 kph to pass a 100. m
long train moving at 100. kph in the a) same direction b) opposite direction
(45 s; 4.1 s)
10. Two cars are at rest, facing each other, and 200. metres apart.
1
2
If car 1 accelerates at 6.00 m/s2 up to maximum speed of 30.0 m/s and car 2
accelerates at 5.00 m/s2 up to a maximum speed of 35.0 m/s, how long does it
take the cars to meet?
(6.08 s)
→ →
11. Prove that: vf 2 - v i 2 = 2 a • d
Physics 12 Student Workbook
©Donald J. Mathewson 2002
26
Unit 1 Lesson 5
Exercises
1. A mass is dropped from a 55.0 m high cliff. Find a) time to fall
b) speed at impact
c) velocity at impact
(3.35 sec; 32.8 m/s - 32.8 ^j m/s)
2. A mass is dropped from a 30.0 m high cliff, When it hits the ground, it stops
in 0.0500 seconds. Find
a) time to fall
b) speed at impact
c) acceleration (and dir'n) during impact
(2.47 s; 24.2 m/s; +485 ^j m/s2)
3. On level ground, a mass is thrown straight up at 25 m/s. If we ignore air
resistance, then find
a) total time in air
b) time to top
c) maximum height
(5.1 s; 2.6 s; 32 m)
4. A projectile is thrown straight up at velocity v. If the acceleration is (-g ^j)
m/s 2 then find an expression for
a) total time in air
b) time to top
c) maximum height
(2v/g; v/g; v 2/2g)
5. A mass is thrown straight up at 25 m/s. At what times will the speed be 18
m/s?
(0.71 sec rising; 4.4 s falling)
6. A student bends her knees 15.0 cm and jumps up 80.0 cm off the ground. Find
a) the velocity with which the student leaves the ground
b) the acceleration of the student while on the ground
(3.96 m/s; 52.3 m/s2)
7. From the edge of a 18.0 m high cliff, a rock is thrown straight up at 25.0 m/s
so that it falls to the base of the cliff. Find the total time of flight (5.74 s)
8. From the edge of a 25 m high cliff, a rock is thrown straight up at 12 m/s so
that it falls to the base of the cliff. Find
a) total flight time
b) speed at impact
c) max height
(3.8 s; 25 m/s; 32 m above base of cliff)
9. From the edge of a 40.0 m high cliff, a rock is thrown straight up at 8.00 m/s
so that it falls to the base of the cliff. Find the time and position of the rock
when its speed is 14.0 m/s.
(2.24 s; 6.73 m below cliff OR 33.3 m above base)
10. From the edge of a 30.0 m high cliff a mass is thrown downwards at 5.00
m/s. Find the time it takes to hit the ground. (2.02 sec)
11. From the top of a 30.m high cliff a rock is dropped. At the same instant from
the base of the cliff, another rock is thrown upwards at 40. m/s. At which time
and location do the rocks pass each other?
(0.75 sec; 2.8 m below cliff edge)
12. A mass is thrown upwards at 29.4 m/s. Plot a graph of velocity versus time
(starting and ending with the object at rest).
13. A runner starts at rest and accelerates up to top speed (v) in 50. metres,
then runs another 50. m at this maximum speed (v). If the time for the race is
15s, find the maximum speed v (10. m/s)
14. A rock is thrown straight up into the sky. At what point is it acceleration
zero? a) as it rises
b) at the top
c) as it falls
d) never
Physics 12 Student Workbook
31
©Donald J. Mathewson 2002
Unit 1 Lesson 6
Exercises
→
1. Given v = 3.0 m/s 20.° S of E
→
→
a) 2 v
b) -v
draw and give the measure of
(6.0m/s at 20°SofE; 3 m/s at 20°NofW)
→
→
2. If
x = 2.00 N SW and y = 3.00 N 10.0° N of W then find:
→ →
→ →
→ →
→ →
a) x + y
b) 2x + y
c) y - x
d) 2x - y
( 4.46 N @ 11.6° SofW; 6.23 N @ 21.8° SofW; 2.47N @ 51.5°NofW; 3.35 N@ 87.8° SofE)
→
A = 3.00 cm at 60.0° N of E
→
C = 6.00 cm at 60.0° S of E
→ → → →
Draw and measure the sum: A + B + C + D
3. Given:
→
4. Add the vectors: d1= 6.0 cm at 30.° N of E
→
d2 = 5.0 cm at 40.° E of S
→
5. Solve for A given that
of E
→ → →
A + B=C ;
→
B = 7.00 cm at 40.0° W of N
→
D = 4.00 cm W
( 4.86 cm at 34.6° N of W)
(8.4 cm at 5.6° S of E)
→
→
B = 10. N 20.° S of E;
C = 8.0 N at 25°S
(2.1 N@ 1.0 ° N ofW)
→
6. An object is moving at vi = 3.0 m/s East . It accelerates for 2.0 seconds at
→
→
→
→
a = 2.0 m/s2 North . a) find vf by adding vi and at
(5.0 m/s at 53° N of E)
→
b) why is v f not 7.0 m/s?
→
→
7. Given vi = 40. m/s E vf = 30. m/s S t = 5.0 s
→
→ →
→
a) find a = (vf - v i ) / t
b) why is a not -2 m/s 2 ?
(10. m/s2 at 38° S of W)
8. Add boat/plane velocity to wind/water velocity to find resultant velocity .
→
→
→
V boat + W = Vresult
a) A boat has an engine speed of 7.00 m/s North. The water current is flowing
at 5.00 m/s West . Find the resulting velocity. ( 8.60 m/s at 35.5° W of N)
b) A boat that moves at 5.00 m/s in still water is heading South. It encounters a
current of 2.00 m/s West. Find the resulting velocity. (5.39 m/s 21.8° W ofS)
c) A plane has an airspeed of 300. kph NE. The wind is blowing at 50.0 kph S.
Find the resulting velocity.
( 267 kph at 37.4° NofE)
d) A plane is heading at 450. kph 25.0° E of N. The wind is blowing at 70.0 kph E.
Find the resulting velocity.
(483 kph at 57.5° N of E)
9. A mass is thrown straight up at 10. m/s. At what times will it be at height
5.0 m off the ground?
(0.88 s; 1.2 s)
Physics 12 Student Workbook
©Donald J. Mathewson 2002
37
Unit 1 Lesson 7
Exercises
^
a) 50.0 m/s at 30.0° N of E (43.3î+25.0j
)
1. Convert to components
^
(19.2î - 16.1j
)
b) 25.0 m/s, 40.0° S of E
(-57.4î -48.2 ^j)
c) 75.0 m at 40.° S of W
2. Convert to polar the following velocities of a projectile
(39.0 m/s, 50.2° above hzntl
25.0 m/s hzntl
39.0 m/s, 50.2° below hztl
51.5 m/s, 60.9° below hzntl)
25.0 i
30.0 j
25.0i
25.0 i
-30.0j
25.0i
-45.0j
→
→
3. Given
A = 20.0 m, 30.0° N of E
B = 10.0 m, 20.0° W of S
a) convert both vectors to rectangular components
b) add the vectors in rectangular
c) convert the answer to polar form
→ →
d) make a rough scale drawing of A + B to confirm your answer to c)
^
^
(17.3î + 10.0j
; -3.42î - 9.40 ^j ; 13.9î + 0.6j
; 13.9 at 2.48° NofE)
→
→
4. Given vi = 10. î + 20. ^j m/s a = -10. ^j m/s2
t = 3.0 s , find in components
→
→ → →
a) the vector at
b) the vector vf = vi + at
^
^
(-30.j
; 10. î - 10.j
)
→
→
5. Given vi = 40. m/s S vf = 30. m/s W t = 5.0 s
→ →
a) write both vectors in rectangular
b) find v f - v i in rectangular
→
→
c) find a in rectangular
d) change to polar a
^
^
(-40.j^; -30.î; -30.î + 40.j
; -6.0î + 8.0j
; 10. m/s2 53° N of W)
Physics 12 Student Workbook
©Donald J. Mathewson 2002
42
→
→
6. Add boat/plane velocity Vboat to wind/water velocity W to find resultant
→
velocity Vresult .
a) A plane travels southeast at 400. kph. The wind is blowing west at 80.0 kph.
Find the plane's velocity relative to the ground.
b) A plane heads 40.° W of N at an airspeed of 380 kph. The plane is seen to
move at 10.°W of N at 350 kph. What is the wind velocity?
a) (283î - 283j^) + (-80î+0j^) = 203î -283 ^j = 348 kph at 54.4° S of E
^
^
b) (-60.8i^ +345 ^j) - (-244i^ +291 ^j) = 183i
+ 54j
= 190kph at 16°NofE
7. Fred and Wilma leave their house at the same time heading for the bus stop
100. m away. Fred starts at rest, accelerates at 0.500 m/s 2 up to a maximum
speed of 5.00 m/s, which he holds as he runs for the bus stop. Wilma gives Fred
a 5.00 second head start and then accelerates for 6.00 seconds up to her
maximum speed of 6.00 m/s. Who gets to the bus stop first, and how long do
they wait for the second person to arrive? (Wilma takes 24.7s; Fred takes 25.0 s)
8. A car starts at rest and accelerates in 7.00 seconds up to a maximum speed of
30.0 m/s, and holds this speed for another 25.0 seconds.
a) find the total distance travelled
(855 m)
b) find the average speed
(26.7 m/s)
9. Make a graph of x-t and v-t for the car in question #8
→ →
→ →
10. By means of a scale drawing find: a) A + B
b) A - B
→
→
A = 100. m, 40.° N of W
B = 80. m, 10.° E of S
(64 m at 13° S of W; 170 m at 58° N of W)
11. Consider the following vectors
→
A = 80.0 m, 20.0° N of W
a) find their components
→ →
b) find A + B
→ →
c) find A - B
Physics 12 Student Workbook
©Donald J. Mathewson 2002
→
B = 55.0 m, SW
(-75.2î + 27.4 j; -38.9î - 38.9 j)
(115. m, 5.77° S of W)
(75.5 m; 61.3° N of W)
43
Unit 1 Lesson 8
Problems
1. A boat heads directly across a current flowing 2.0m/s South. the boat is
moving at 5.0 m/s towards the East. Find the resultant velocity (and direction)
relative to the ground.
(5.4 m/s at 22°SofE)
2. A boat with an engine speed of 10. kph is moving straight across a river
with a 8.0 kph current
a) At what angle is the boat pointing?
(53° upstream)
b) What is the boats resultant speed?
( 6.0 kph)
3. Draw a precise vector diagram for the following
a) a plane flies at 350. m/s at 25° Nof E and the wind blows at 120. m/s South.
b) A boat heads directly across a 4.0 m/s current at a speed of 7.0 m/s
c) A plane flies at 250 m/s at 35° Nof E and the wind is 90.m/s at 70° E of S
d) a plane has a resultant velocity of 550 m/s NE and airspeed 350 m/s East
e) A person wants to cross a 60m wide river (current 3.0m/s) in 12 seconds.
(assume the swimmer crosses as quickly as possible)
f) a boat wants to go at 40. m/s and 25° SofE in a current of 25 m/s at 35°NofE
4. In 3a) find the resultant velocity (& dir'n) for the plane: a plane flies at 350.
m/s at 25° Nof E and the wind blows at 120. m/s South
(320 m/s at 5.0° N of E)
5. In 3b) find the resulting speed (& dir'n) for the boat: A boat heads directly
across a 4.0 m/s current at a speed of 7.0 m/s
(8.1 m/s at 30.° downstream)
6. In 3c) find the resulting velocity (& dir'n) for the plane: A plane flies at 250
m/s at 35° Nof E and the wind is 90.m/s at 70.° E of S
(310 m/s at 21° N of E)
7. In 3d) find the velocity of the wind (& dir'n): a plane has a resultant velocity
of 550 m/s NE and airspeed 350 m/s East
(390 m/s at 84° N of E)
8. In 3e) find the swimmer speed (& dir'n): A person wants to cross a 60. m wide
river (current 3.0m/s) in 12 seconds.
(5.8 m/s at 31° downstream)
9. In 3f) find the boat speed and dir'n: a boat wants to go at 40. m/s and 25° SofE
in a current of 25 m/s at 35°NofE
(35 m/s at 63° S of E)
10. The current in a 25.0 m wide river is 2.00 m/s towards the East. If you swim
South at 5.00 m/s across the river
a) what is your velocity with respect to shore?
b) how long does it take you to cross the river?
c) how far downstream do you reach the other shore?
( Vr = 5.38 m/s 21.8° E of S; t = 5.00s; 10.0m)
Physics 12 Student Workbook
©Donald J. Mathewson 2002
48
11. The current in a 30.0 m wide river is 3.00 m/s towards the North. If you
need to move straight across the river and you can swim at 4.00 m/s
a) in what direction should you head
b) how long does it take you to cross the river?
c) how far downstream do you reach the other shore?
( θ = 48.6° upstream; 11.3 sec; 0metres)
12. If you swim at 4.00 m/s and 70.0° upstream and in this way you move
straight across a river:
a) what is the speed of the current?
b) what is your speed relative to the ground?
( W = 3.76 m/s Vr = 1.37 m/s)
13. A plane flies north at 200. kph. A wind is blowing towards the northeast at
100. kph
a) what is the plane's velocity with respect to the ground?
b) in one hour, how far off course will the plane be?
c) to stay on course, what direction should the pilot head relative to the air?
(280. kph 75.4° N of E; 70.7 km off course; 20.7° W of N)
14. A plane flies 200. kph SW relative to the air. A radar on the ground sees the
plane moving at 180. kph 30.0° S of W.
a) what is the wind velocity in rectangular?
b) what is the wind speed?
c) what is the wind direction?
^
(W = -14.5 î + 51.4 j = 53.4 kph at 74.3° N of W)
15. A plane has an airspeed of 300. kph and is heading 15.0° E of N. A wind
blows towards the east such that the plane moves 25.0° E of N relative to the
ground.
a) what is the windspeed?
b) what is the groundspeed?
( W = 57.5 kph; Vr = 320. kph)
16. The current in a 25.0 m wide river is 2.00 m/s towards the South. If you can
swim at 4.00 m/s in still water
a) how long does it take you to cross if you swim northeast relative to the water?
b) what is the fastest time for crossing the river?
(8.84 sec ; 6.25 s(directly across))
17. A plane has an airspeed of 300 kph and its heading vector lies in quadrant
III. A 100 kph wind is blowing such that the plane moves at 300 kph S with
respect to the ground.
a) what is the plane's heading?
b) what is the wind direction?
Physics 12 Student Workbook
©Donald J. Mathewson 2002
(19.2° W of S)
(9.59° S of E)
49
Unit 1 Lesson 9
Projectile Exercises
1. Find the flight time and range for each of the following projectiles
v
h
a) h = 39.2 m and v = 100. m/s
b) h=39.2 m and v = 200. m/s
c) h=88.2 m and v = 100. m/s
d) h = 88.2 m and v = 200. m/s
(2.82 s, 282 m)
(2.82 s, 566 m)
(4.24 s, 424 m)
(4.24 s, 849 m)
2. Find the flight time and range for these projectiles
v
θ
a) v = 100. m/s and θ= 45.0°
b) v = 100. m/s and θ = 60.0°
c) v = 200. m/s and θ= 45.0°
(14.4 s, 1020m)
(17.7 s, 884 m)
(28.8 s, 4080 m)
3. Find the flight time and range of the following projectiles
v
θ
h
a) h = 14.7 m v= 19.6 m/s and θ = 30.0° (3.00s, 50.9m)
b) h = 78.4 m v = 41.6 m/s and θ = 45.0° (8.00s, 235m)
4. If a football player kicks a football so that it spends 3.0 seconds in the air and
travels 50. m, then find the velocity and angle of the kick.
(note: picture is like #2 above)
(22 m/s, 41°)
Physics 12 Student Workbook
©Donald J. Mathewson 2002
57
5. For the projectile pictured in #3 a) find
a) the horizontal velocity after 1.30 seconds
b) the vertical velocity after 1.30 seconds
c) the resultant velocity after 1.30 seconds
(17.0m/s, -2.94m/s, 17.2 m/s at 9.83° below hzntl)
6. For the projectile in #2a) find
a) the horizontal position/displacement after 3.00 seconds
b) the vertical position after 3.00 seconds
c) the resultant displacement after 3.00 sec
(212 m, 168 m, 271m at 38.4° from launch point)
7. For the projectile in #1a) find the resultant velocity (& angle) just before
the projectile hits the ground (104 m/s at 15.5° below hzntl)
8. A projectile is launched on level ground. Two seconds later it is moving at
30.0 m/s at 20.0° S of E. Find its launch velocity and direction
(29.7 m/s at 18.3°above hzntl)
9. The horizontal and vertical speed of a projectile are measured in an
experiment. The data is below:
time (sec) 0.0 s
0.1 s
0.2 s
0.3 s
0.4 s
0.5 s
0.6 s
x-speed(m/s) 2.0
2.0
2.0
1.9
1.9
1.9
1.8
y-speed(m/s) 0.0
0.9
2.1
2.9
4.1
5.0
5.8
a) graph the horizontal velocity versus time and draw conclusions
b) graph the vertical velocity versus time and draw conclusions
10. A projectile is launched at 30. m/s and 35° on level ground. When it has
travelled horizontally 14 metres, what is its speed and direction?
(27 m/s at 25° above hzntl)
11. A projectile is launched on level ground at 25.0 m/s and at 35.0°. What is its
maximum height?
(10.5 m)
12. A projectile is launched at 35.0 m/s and at 30.0° on an 20.0° angled slope.
Where does it land (distance along slope measured from launch) (42.6 m)
30.0°
20.0°
Physics 12 Student Workbook
©Donald J. Mathewson 2002
58
Unit 1 Lesson 10
Exercises and Review
1. A projectile is launched from a 80.0 m high cliff at a velocity of 30.0 m/s and
an angle of 40.0° above the horizontal. Find its
a) position (x and y) after 3.50 sec
(80.4 m right, 7.47m up)
b) velocity (resultant) after 3.50 sec
(27.4 m/s at 33.2° below hzntl)
2. A projectile is launched at an angle on level ground. Which of these graphs
is a graph of its
a) horizontal displacement vs time
b) vertical displacement vs time
c) vertical velocity versus time d) horizontal velocity versus time
ii
i
iii
iv
3. Find the speed and direction of the projectile when it hits the wall.
25 m/s
28°
45.0 m
(24 m/s; 20.° below hzntl)
4. Write the equations you would use to find the launch speed of the projectile
shown below if it were to land at the point marked x.
x
y
40.0 m
v
40.0°
30.0 m
160. m
(160 = v cos(40) t; 30 = v sin(40) t - 4.9 t2)
5. For the diagram above, find the launch speed of the projectile if it lands
exactly half-way between x and y.
(47.3 m/s)
6. A projectile is launched horizontally off of a cliff. When it hits the ground at
the base of the cliff, it is moving at 50.0 m/s and 20.0° below hzntl. Find the
cliff height and range of the projectile.
(14.9m; 82.0m)
7. A person walks 30.0 m at 20.0°North of East at 2.00 m/s, then 5.00 m East at
1.00 m/s, then 2.00m North at 0.500 m/s
a) how much time elapsed?
(24 s)
b) what is the resultant displacement?
(35.4 m at 20.3° N of E)
c) what is the average velocity?
(1.47 m/s at 20.3° N of E)
Physics 12 Student Workbook
©Donald J. Mathewson 2002
64
8. A boat heads directly across a 4.0 m/s current moving at an engine speed of
6.0 m/s. When it reaches the other side of the 30.m wide river
a) how much time has elapsed?
(5.0 seconds)
b) how far downstream is the boat?
(20. m)
c) what resultant distance has it travelled?
(36 m)
9. A projectile is launched on level ground at 20. m/s and 30.° above the hzntl.
→ → →
Find the velocity after 0.50 seconds by graphical vector addition: vf = vi + at
(18 m/s at 16° above hzntl)
10. The engineer of a passenger train moving at 29.0 m/s sees a freight train
360.m ahead on the track moving at 6.00 m/s. The engineer takes 0.400
seconds to react, and then applies the brakes. At what rate must the
passenger train then decelerate in order to avoid a collision? (0.754 m/s2)
for 11-20 answers on page 66:
11. A bullet fired horizontally hits the ground in 0.5 sec. If it is fired in the
same direction with higher speed, how long will it be in the air?
a) t > 0.5 s
b) 0.5 s
c) t < 0.5 s
12. Neglecting air resistance, a horizontally fired projectile maintains its
horizontal component of speed because
a) it is not acted upon by any force
b) it is not acted upon by any horizontal forces
c) it has no vertical component of speed to begin with
d) the net force acting on it is zero
13. A hunter aims a rifle at 10° above the horizontal and fires. At the same
instant, the hunter drops a bullet from his hand while holding his rifle. The
bullet to hit the ground first will be
a) dropped b) fired
c) both hit at same time
14. If an object is dropped, it accelerates down at 1g. If instead it is thrown
upwards and follows a parabolic path, its acceleration in flight is
a) 0
b) 0< a < 1 g
c) 1g
d) a> 1g
15. A projectile is fired on level ground at 50° and hits the ground down range.
It will land in the same place if it is instead fired at
a) 40°
b) 45°
c) 55°
d) 60°
16. A rifle with muzzle velocity 100m/s is fired horizontally from a tower. After
one second, how far will the bullet have travelled horizontally?
a) 50 m
b) 100m c) > 100m
17. Two projectiles are fired on level ground with the same speed. One is fired
at 30° and the other at 60°. Which projectile hits the ground first?
a) 30°
b) 60°
c) hit at same time
18. An object is thrown straight into the air. If we include air resistance, then
which is greater: the time for the ascent(rising) or the time for the descent
(falling)
a) descent
b) ascent
c) same
Physics 12 Student Workbook
©Donald J. Mathewson 2002
65
19. A baseball is thrown horizontally from a height of 5 metres and it lands
20m down range. The pitching speed is approximately
a) 5 m/s
b) 10 m/s
c) 20 m/s
d) 25 m/s
20. An airplane flying horizontally at 400m/s at an altitude of 500m drops a
package. How far does the package travel horizontally before landing?
a) 0 km, it lands right underneath the release point
b) 4 km
c) 5 km
d) > 5km
21. For a projectile launched horizontally off of a cliff, if the launch speed
doubles, what happens to the range?
a) no change
b) doubles
c) halved
d) increased by 2
e) quadrupled
22. For a projectile launched horizontally off of a cliff, if the cliff height
doubles, what happens to the range?
a) no change
b) doubles
c) halved
d) increased by 2
e) quadrupled
23. For a projectile launched at an angle on level ground, if the launch speed
doubles, what happens to the the range?
a) no change
b) doubles
c) halved
d) increased by 2
e) quadrupled
24. Two projectiles are launched on level ground at equal speeds, one at 30°,
the other at 60°. Which answer best explains why they travel the same
horizontal distance?
a) the 60° projectile spends less time in the air
b) their vertical distances are the same, so their horizontal distances are also
c) the 30° projectile and the 60° projectile have the same flight time
d) the 30° projectile has less flight time, but more horizontal speed
e) the 30° projectile has a greater speed than the 60° projectile
25. A plane travelling at 100 m/s drops a package from a height of 500 m. When
the package hits the ground, what is the distance from the plane to the
package?
a) 500 m
b) 1000 m
c) 1120 m
d) > 1120 m
11
b
12
b
13
a
14
c
15
a
16
b
Physics 12 Student Workbook
©Donald J. Mathewson 2002
17
a
18
a
19
c
20
b
21
b
22
d
23
e
24
d
25
a
66
Lesson 11: Practice Exam
Section I Multiple Choice
30 questions, 1 marks each for 30 marks total.
1. Which of the following is true for this projectile?
a) the distance travelled is zero
b) the vertical displacement is zero
c) the position is zero
d) the initial velocity is zero
e) the horizontal displacement is zero
2.
a)
c)
e)
Which of the following is the best definition of acceleration?
the rate of change of velocity
b) the rate of change of position
velocity/time
d) displacement/time
change in velocity
3. A car moving at 20 m/s decelerates at a rate of 4.0 m/s2 when the brakes are
applied. How far does the car travel before stopping?
a) 5 m
b) 0.2 m
c) 25 m
d) 100 m
e) 50 m
4. If X is 3 m East and Y is 4m South, then find X + Y
a) 5 m Southeast
b) 7 m Northeast
d) 5 m 37° S of E
e) 5 m 53° N of E
c) 5 m 53° S of E
5. If X is 3 m East and Y is 4m South, then find X - Y
a) 5 m Southeast
b) 7 m Northeast
d) 5 m 37° S of E
e) 5 m 53° N of E
c) 5 m 53° S of E
6. If vector A points N
vector B points E
and vector C points S
then which of the following is A - B + C?
a)
b)
c)
d)
e)
7. Change to polar the vector F = (-2.5 î + 4.0 ^j ) N
a) 1.5 N; 32° N of W
b) 4.7 N; 32° N of W
c) 22.5 N; 58° N of E
d) 4.7 N; 58° N of E
e) 4.7 N; 58° N of W
8. Change to rectangular components the vector: a =
^
^
a) -6.19i - 5.97 j
d) -5.97 ^i + 6.19 ^j
^
^
b) 6.19i - 5.97 j
e) -6.19 ^i + 5.97 ^j
Physics 12 Student Workbook
©Donald J. Mathewson 2002
8.60 m/s2 at 44° S of W
^
c) -5.97i^ - 6.19j
67
9. If the initial velocity of an object is Eastward and the final velocity of the
object is towards the South , then what is the direction of the acceleration?
v
i
v
f
a) south
b) northwest
c) southwest
d) southeast
e) northeast
10. You head directly across a river that is flowing at 4.0 m/s. If you can
swim at 2.0 m/s, then what is your velocity relative to shore?
N
current
E
heading
a) 4.5 m/s at 63° E of N
d) 3.5 m/s at 27° N of E
b) 6 m/s E
e) 4.5 m/s at 27° E of N
c) 3.5 m/s at 63° E of S
11. A kayaker has to move straight across a 20. m wide river that is flowing at
2.5 m/s. If the kayaker can paddle at 3.5 m/s, then at what angle should they
head, and how long does it take them to reach the other shore?
N
current
E
kayak
waterfall
20 metres
a) 44° N of E
b) 46° N of E
c) 55° N of E
d) 44° N of E
e) 46° N of E
time=
time=
time=
time=
time=
4,6 s
8.2 s
1.1 s
8.2 s
4.6 s
12. A boat heads directly across a 30. m wide river at a speed of 3.5 m/s. If the
current in the river is 1.5m/s then how far downstream is the boat when it
reaches the other shore?
a) 25.8 m
b) 8.6 m
c) 17.2 m
d) 70 m
e) 12.9 m
13. A plane flies North at 300 kph. A wind is blowing from the west such that
the groundspeed is 330 kph. What is the direction of the velocity relative to the
ground?
a) 25° W of N
b) 65 ° E of N
d) 43° W of N
e) 25° E of N
Physics 12 Student Workbook
©Donald J. Mathewson 2002
c) 43° E of N
68
14. A plane flies with an airspeed of 350. kph east. The wind is blowing at 150.
kph at 60° N of W. Find the resultant ground speed and direction.
a) 381 kph, 23° N of E
b) 444 kph, 17° N of E
c) 242 kph, 18° N of E
d) 304 kph, 25° N of E
e) 200 kph, NE
15. A ball rolls off the top of a 10.0 m high building at a speed of 3.00 m/s. How
long does it take to hit the ground?
v = 3 m/s
10m
a) 3.33 s
b) 2.00 s
c) 1.43 s
d) 0.71 s
e) 1.00 s
16. In question 15, how far from the base of the building does the ball land?
a) 3.0 m
b) 9.9 m
c) 4.3 m
d) 6.0 m
e) 29.4 m
17. In question 15, if the speed of the ball is doubled, what happens to the
range R ? (its distance from the base of the wall)
a) R is halved (× 0.5)
b) R doubles (× 2) c) R is unchanged
d) R is quadrupled (× 4)
e) R increases by 2 (× 2 )
18. In question 15, what is the speed (resultant velocity magnitude) of the ball
just before it hits the ground?
a) 14.6 m/s
b) 14.3 m/s
c) 14.0 m/s
d) 29.4 m/s
e) 16.7 m/s
19. A projectile is launched at an angle on level ground. Which of the
following is true about the resultant speed and the acceleration of the
projectile at the top of its trajectory?
speed
a) positive
b)
0
c)
0
d) positive
e) constant
acceleration
-9.8
-9.8
0
0
0
Physics 12 Student Workbook
©Donald J. Mathewson 2002
69
20. A projectile is launched on level ground as in question 19. The launch
speed is 42.0 m/s and the launch angle is 25.0°:
42.0 m/s
25.0°
Find the vertical component of the initial velocity.
a) 42.0 m/s b) 5.90 m/s c) 18.6 m/s d) 17.7 m/s
e) 38.1 m/s
21. In question 20, find the speed of the projectile at the top of its trajectory.
a) 42.0 m/s b) 5.90 m/s c) 18.6 m/s d) 17.7 m/s e) 38.1 m/s
22. In question 20, find the flight time of the projectile.
a) 1.81 s
b) 3.62 s
c) 7.77 s
d) 3.92 s
e) 8.57 s
23. In question 20, find the range of the projectile, R.
a) 67.3 m
b) 163 m
c) 138 m
d) 68.9 m
e) 149 m
24. Find the maximum height of this projectile above the base of the cliff.
19.6 m/s
30°
39.2 m
a) 19.6 m
b) 4.90 m
c) 39.2 m
d) 58.8 m
e) 44.1 m
25. Find the flight time of the projectile in question 24.
a) 1 s
b) 2 s
c) 4 s
d) 3 s
e) 3.5 s
26. A plane flying horizontally at speed v drops a package. After 3 seconds,
what is the speed of the package in m/s ? Ignore air resistance.
a) 29.4
b) v + 29.4
c) v
d)
v 2+29.42
e) v - 29.4
27. A projectile is launched at speed v and angle θ >45° on flat ground. If the
speed v is kept constant and the angle is increased slightly, then what
happens to the flight time and range?
time
range
a
no change
no change
b
increases
decreases
c
increases
increases
d
decreases
no change
e
decreases
decreases
Physics 12 Student Workbook
©Donald J. Mathewson 2002
70
28. Which of the following best describes projectile
horizontal motion
a
constant velocity
b
zero acceleration
c
constant velocity
d
zero velocity
e
zero acceleration
motion?
vertical motion
constant acceleration
constant velocity
zero velocity
constant acceleration
zero acceleration
29. A projectile is launched at 40 m/s and 30° above the horizontal from a cliff.
At what time during the flight will the speed be 30 m/s?
a) 2.0 s
b) 1.0 s
c) 4.1 s
d) 3.1 s
e) never
30. A projectile is launched at 40 m/s and 30° above the horizontal from a cliff.
At what time during the flight will the speed be 45 m/s?
a) 5.0 s
b) 0.5 s
c) 4.1 s
d) 2.9 s
e) never
1
b
2
a
3
e
4
c
5
e
6
d
7
e
8
a
9
c
10
a
11
b
12
e
13
e
14
d
15
c
16
c
17
b
18
b
19
a
20
d
21
e
22
b
23
c
24
e
25
c
26
d
27
b
28
a
29
e
30
a
Physics 12 Student Workbook
©Donald J. Mathewson 2002
71
Section II Problems
20 marks total. 5 marks each
1. A plane has an airspeed of 410. kph in a direction 40.0° S of E. It encounters
a northerly (towards the north) wind and its resultant motion is along 25.0° S
of E. Find:
a) the speed of the wind b) the resultant speed
(117 kph; 346 kph)
y
2. For the following projectile, find:
a) the polar velocity after 2.0 s
70. m
20. m/s
40.°
b) the position (x and y) after 2.0s.
0m
x
(17 m/s at 24° below hzntl; (31î + 76j^ ) metres)
3. a) Sketch a stop action picture (3-4 frames) for the motion of the projectile
below. Include velocity components on your picture.
b) explain by referring to forces why the velocity changes in this way.
4. Joe Carter hits a baseball and it leaves the bat at an angle of 35.0° and a
height of 1.00 m above the ground. The ball travels 100.m and barely passes
over a 12.0 m high fence. At what speed did the ball leave the bat?
home run!
Physics 12 Student Workbook
©Donald J. Mathewson 2002
(35.2 m/s)
72
other (hard) practice exam questions
1. A projectile is launched off of a 30.m high cliff at 25° and 40. m/s. the base
of the cliff is irregular as shown in the diagram. Where exactly (x and y) does
the projectile land?
(x=160m,y=9.2m ie 0.8m from top of 10 m wall)
40 m/s
25°
30 m
y
10 m
160 m
x
2. A gun can shoot charges at 250 m/s. The gun is 2500 m from the base of a
1800 m high mountain. For some angles, the shot will hit the mountain and not
reach the enemy. Your task is to find the range of angles for which the gun
will be able to clear the mountain (eg. from 10° to 15°).
1.8 km
gun
θ
2.5 km
mountain
(50° to 76°)
3. At amusement parks, there is often a contest involving throwing softballs
into milk cans. For the softballs to go in, they must enter at an angle of 80.0°
below the horizontal. The top of the milk cans are 200. cm horizontally and
+10.0 cm vertically from the thrower. At what angle and speed should the
thrower release the softball for it to go in?
v
80°
10 cm
200 cm
(7.66 m/s; 80.2° N of E)
Physics 12 Student Workbook
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73