Two Dimensional
Kinematics
Position and Velocity Vectors
y
z
A
y
x
z
x
If an object starts out at the origin and moves to point A, its
displacement can be represented by a position vector.
x
+ y
+ z
Position and Velocity Vectors
As an object moves from one point in space to another, the average velocity of its motion can
be described as the displacement of the object over the time it takes to move.
(average velocity vector)
To find the instantaneous velocity (the velocity at a specific point in time) it requires the time
interval to be so small that it can effectively be reduced to 0 which can be represented as a
limit expression.
(instantaneous velocity vector)
Components of
Instantaneous Velocity
The instantaneous velocity can have three different components: x, y,
and z.
Each component is shown below,
Vector representation:
Acceleration Vector
Acceleration is the rate at which the velocity is changing, and the average acceleration
can be found by taking the difference of the final and initial velocity and dividing it by
the time it takes for that event to occur.
Just as we can find the velocity at a specific point in time, we can also find
the instantaneous acceleration using a limit expression.
Components of
Instantaneous Acceleration
Vector representation:
Given
and .
. Find
Given . Find and .
Projectile Motion
Have you ever thrown an object in the air and watched the trajectory it
follows?
The path the object travels is a parabolic path.
vy
vx
vx
vy
vx
v
vy
vx
vx
vy
Velocity of a Projectile
vy
vx
vx
vy
vx
v
vy
vx
vx
vy
To explain projectile motion in the vertical direction we can use our knowledge of
throwing a ball straight up into the air. We know that eventually the acceleration
due to gravity will eventually stop the ball and make it move back towards the
Earth. At its apex the ball stops moving in the vertical direction, so for a
projectile this would be the same.
Velocity of a Projectile
vy
vx
vx
vy
vx
v
vy
vx
vx
vy
To account for the projectile's motion in the horizontal direction we imagine a case
where a block moves to the right with a velocity, v. In the absence of a resistant force
(e.g. air resistance or friction) we can state that the block moves with a constant
velocity (note that gravity does not affect the projectile's horizontal motion).
Position of an object in projectile
motion
The position of the projectile with respect to its starting position can be
represented with minor changes to the kinematics equation:
Horizontal position:
Vertical Position:
From these equations we can determine the maximum horizontal distance, the
maximum height reached by the projectile, the time to reach its highest point,
and the time it hits the floor again.
1
A
Which of the following statements are true regarding
projectile motion?
is constant
B
Acceleration is +g when the object is rising and -g when falling.
C
In the absence of friction the trajectory will depend on the object's
mass as well as its initial and launch angle.
D
The velocity of the object is zero at the point of
maximum elevation.
E
The horizontal motion is independent of the vertical motion.
2
A marble is shot and follows a parabolic path shown below. Air
resistance is negligible. Point Y is the highest point on the path.
Which of these indicates the direction of the speed, if any, of the marble
at point Y?
A
B
v
C
D
E
None
3
A marble is show and follows a parabolic path shown above.
Air resistance is negligible. Point Y is the highest point on the
path.
Which of the following indicates the direction of the net
force on the marble at Point X?
A
B
C
D
E
v
Time to fall from apex
When a projectile is thrown in the horizontal direction
v
H
4
Two cannon balls are launched simultaneously off a cliff. The
two cannon balls have different masses and different initial
velocities. Which will strike the ground first?
A
The heaviest one
B
The lightest one
C
The slowest one
D
The fastest one
E
They will both strike the ground at the same time
To Find Maximum Height
Because at the highest point the vertical
component of velocity is zero.
(time to attain maximum height)
To Find Maximum Displacement
vy
vx
vx
vy
v
vx
vy
vx
vx
vy
To find angle between the velocities
vx
vy
vy
vx
vy
vy
5
At what angle will a projectile have the greatest vertical
displacement?
A
0
B
30
C
45
D
60
E
90
At what angle will a projectile have the greatest horizontal
displacement?
6
A
0
B
30
C
45
D
60
E
90
7
Which angles will have the same horizontal displacement?
A
0 and 90
B
30 and 60
C
0 and 45
D
35 and 60
E
30 and 90
F
None of the two angles above will have the same
displacement.
Moving in a Circular Path
constant speed
decreasing speed
increasing speed
When an object moves in a circle with constant speed and its acceleration is
perpendicular to the velocity this is called Uniform Circular Motion.
8
A car is driving with decreasing velocity on a curved path. Which
diagram shows the correct direction for the velocity and
acceleration?
v
A
v
B
a
a
C
D
v
v
a
a
E
v
a
9
A car is driving with constant velocity on a curved path. Which
diagram shows the correct direction for the velocity and
acceleration?
v
A
B
v
a
a
C
D
v
v
a
a
E
v
a
10
A car is driving with increasing velocity on a curved path. Which
diagram shows the correct direction for the velocity and
acceleration?
v
A
v
B
a
a
C
D
v
v
a
a
E
v
a
Uniform Circular Motion
(centripetal acceleration)
If we plug the equation for the velocity
into the acceleration equation we get:
Uniform Circular Motion
Centripetal Acceleration
P2
P1
Knowing that the triangles are similar, we can use ratios of corresponding sides, therefo
To find the instantaneous velocity, we first have to come up with a representation
for the average acceleration as before
Uniform Circular Motion
Centripetal Acceleration
To find the instantaneous acceleration we have to take a limit expression of the
average acceleration.
The limit expression will give us the velocity at
a certain point in time, this velocity is the same
as v1
(centripetal acceleration)
11
If a ball is swung in a circle of a radius of 1 m with a velocity of 5
m/s what would be the centripetal acceleration?
A
5 m/s2
B
0.2 m/s2
C
25 m/s2
D
0.04 m/s2
E
10 m/s2
12
If a ball is swung in a circle of radius 9 m and its centripetal
acceleration was 1 m/s2. What would be its velocity?
A
3 m/s
B
9 m/s
C
81 m/s
D
18 m/s
E
√3 m/s
13
If an object is moving in a circle with a velocity of 15 m/s and has
a centripetal acceleration of 45 m/s2. What would be its radius?
A
5m
B
1/3 m
C
3m
D
10 m
E
15 m
Non-Uniform Circular Motion
arad
atan
arad
arad
atan
arad
arad
atan
arad
atan
When you are on a roller coaster
and you come to a circular loop,
your velocity is not constant.
As you approach the top of the
loop your velocity decreases and
as you come back down your
velocity increases. You still
have a radial acceleration but
now there is a tangential
acceleration which is
perpendicular to your radial
acceleration.
Relative Velocity
When you are riding in a car and you look out a window what do you
see?
If there is another car moving along side you with the same velocity
relative to you, the other car appears to stand still, but with respect to
the ground both of you are moving.
If another car is moving with velocity 2v with respect to the ground, then
with respect to your car its moving with a velocity of v.
Relative Velocity
If a plane is flying through the air and enters a crosswind it will have a velocity
straight and one perpendicular to it.
Vplane
Vcrosswind
14
A plane is moving with a constant speed of 1200 km/h and
during part of its flight there is a cross wind blowing at 500
km/h. What is the net velocity during this portion of its flight?
A
1600 km/h
B
1300 km/h
C
700 km/h
D
1700 km/h
E
2500 km/h
1200 km/h
500 km/h
15
Two kids are on a boat capable of a maximum speed of 10
kilometers per hour in water, and wish to cross a river 2
kilometers wide to a point directly across from their starting
point. If the speed of the water in the river is 9 kilometers per
hour how much time is required for the crossing?
A
0.05 hrs
B
0.45 hrs
C
1 hr
D
10 hrs
E
Not possible