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Physics 215
Lecture 2-2
Physics 215 – Fall 2016
02-2
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•  Last time:
–  Displacement, velocity, graphs
•  Today:
–  Constant acceleration, free fall
Physics 215 – Fall 2016
02-2
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2-2.1: An object moves with constant acceleration, starting from
rest at t = 0 s. In the first four seconds, it travels 10 cm.
What will be the displacement of the object in the following four
seconds (i.e. between t = 4 s and t = 8 s)?
A. 
B.
C. 
D. 
10 cm
20 cm
30 cm
40 cm
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02-2
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Motion with constant acceleration:
v = vi + at
vav = (1/2) (vi + v)
x = xi + vit + (1/2) a t2
v2 = vi2 + 2a (x - xi)
*where xi, vi refer to time = 0 s ;
x, v to time t
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Sample problem
•  A brick is dropped (zero initial speed) from the roof of a building. The brick
strikes the ground in 2.50 s. Ignore air resistance, so the brick is in free fall.
–  How tall, in meters, is the building?
–  What is the magnitude of the brick s velocity just before it reaches the ground?
–  Sketch a(t), v(t), and y(t) for the motion of the brick.
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Sample problem
•  A student standing on the ground throws a ball straight up. The ball leaves
the student’s hand with a speed of 15 m/s when the hand is 2.0 m above
the ground. How long is the ball in the air before it hits the ground?
–  Assume that the student moves her hand out of the way.
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Sample problem
•  A cheetah spots a Thomson’s gazelle, its preferred prey, and leaps into
action, quickly accelerating to its top speed of 30 m/s, the highest of any
land animal. However, a cheetah can maintain this extreme speed for only
15 s before having to let up. The cheetah is 170 m from the gazelle as it
reaches top speed, and the gazelle sees the cheetah at just this instant.
With negligible reaction time, the gazelle heads directly away from the
cheetah, accelerating at 4.6 m/s2 for 5.0 s, then running at constant speed.
Does the gazelle escape? If so, by what distance is the gazelle in front
when the cheetah gives up?
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Sample problem
•  A sprinter can accelerate with constant acceleration for 4.0 s before
reaching top speed. He can run the 100 meter dash in 10.0 s. What is his
speed as he crosses the finish line?
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Motion in more than 1 dimension
•  Have seen that 1D kinematics is written
in terms of quantities with a magnitude
and a sign
•  Examples of 1D vectors
•  To extend to d > 1, we need a more
general definition of vector
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Vectors: basic properties
•  are used to denote quantities that have
magnitude and direction
•  can be added and subtracted
•  can be multiplied or divided by a number
•  can be manipulated graphically (i.e., by
drawing them out) or algebraically (by
considering components)
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Vectors: examples and properties
•  Some vectors we will encounter:
position, velocity, force
•  Vectors commonly denoted by boldface
letters, or sometimes arrow on top
•  Magnitude of A is written |A|, or no
boldface and no absolute value signs
•  Some quantities which are not vectors:
temperature, pressure, volume ….
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Drawing a vector
•  A vector is represented graphically by a
line with an arrow on one end.
•  Length of line gives the magnitude of
the vector.
•  Orientation of line and sense of arrow
give the direction of the vector.
•  Location of vector in space does not
matter -- two vectors with the same
magnitude and direction are equivalent,
independent of their location
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Adding vectors
To add vector B to vector A
•  Draw vector A
•  Draw vector B with its tail
starting from the tip of A
•  The sum vector A+B is the
vector drawn from the tail of
vector A to the tip of vector B.
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Multiplying vectors by a number
A
•  Direction of vector not affected
(care with negative numbers –
see below)
•  Magnitude (length) scaled, e.g.
1x
A
=
–  1*A=A
–  2*A is given by arrow of twice
length, but same direction
–  0*A = 0 null vector
2x
0x
–  -A = -1*A is arrow of same length,
-1 x
but reversed in direction
Physics 215 – Fall 2016
02-2
A
A
A
=
=
=
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2-2.2:
Which of the vectors in the second row shows
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?
Slide 3-20
QuickCheck
3.1
2-2.2:
Which of the vectors in the second row shows
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?
Slide 3-21
QuickCheck
3.2
2-2.3:
Which of the vectors in the second row shows 2 -
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?
Slide 3-27
QuickCheck
3.2
Clicker 2-2.3:
Which of the vectors in the second row shows 2 -
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?
Slide 3-28
Projection of a vector
How much a vector acts along
some arbitrary direction
Component of a vector
Projection onto one of the
coordinate axes (x, y, z)
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Components
A = Ax + Ay
y
A
A = axi + ayj
j
x
i = unit vector in x direction
i
j = unit vector in y direction
Projection of A along coordinate axes
ax, ay = components of vector A
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More on vector components
y
•  Relate components to
direction (2D):
ax = |A|cosθ,
ay = |A|sinθ
A
j
θ
i
or
Ay
x
Ax
•  Direction: tanθ = ay/ax
•  Magnitude: |A|2 = ax2 + ay2
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2-2.4: A bird is flying along a straight line in a
direction somewhere East of North. After the
bird has flown a distance of 2.5 miles, it is 2
miles North of where it started.
How far to the East is it from its starting point?
A. 
B. 
C.
D.
0 miles
0.5 miles
1.0 mile
1.5 miles
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Why are components useful?
•  Addition: just add components
e.g. if C = A + B
cx = ax + bx; cy = ay + by
•  Subtraction similar
y
B
j
A
•  Multiplying a vector by a
number – just multiply
components: if D = n*A
x
i
dx = n*ax; dy = n*ay
•  Even more useful in 3 (or
higher) dimensions
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Sample problem
Sam leaves his house and follows the following three step path:
He heads 50.0 m due east. Then he travels 20.0m at 480 north
of east. He then travels 70.0m 620 north of west.
Sketch the graph of Sam’s path in two-dimensions.
What is Sam’s net displacement?
What is the total distance that Sam travels?
If Sam moves at constant speed and the trip takes 100s, what is
Sam’s speed?
What is the magnitude of Sam’s average velocity?
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2D Motion in components
Note: component of position vector along x-direction is the x-coordinate!
y
Q
yQ
s – vector position → s = xQi + yQj
j
i
x
xQ
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Displacement in 2D Motion
y
Δs
sI
sF
s – vector position
Displacement Δs = sF - sI, also a vector!
O
x
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02-2
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