Physics 2514
Lecture 7
P. Gutierrez
Department of Physics & Astronomy
University of Oklahoma
Physics 2514 – p. 1/19
Goals
First we finish the brief introduction to gravity. Then a brief
review (introduction) to the mathematical properties of vectors is
given.
Physics 2514 – p. 2/19
Gravity
In the 17th century Galileo carried out a series of experiments
that found that all objects independent of mass fall toward the
Earth at the same rate.
The rate at which objects fall is g = 9.8 m/s2 (toward the
Earth). This assumes that we neglect resistance due to the
Earth’s atmosphere, and that we are close to the Earth’s
surface.
The value of g is only correct near the surface of the
Earth.
The value of g is different for other astronomical objects
(Earth’s moon g = 1.6 m/s2 , for Mars g = 3.7 m/s2 , for
Jupiter g ≈ 26 m/s2 ).
Physics 2514 – p. 3/19
Gravity
The acceleration toward the Earth is given by ay = −g with
g = 9.8 m/s2
Where we have assumed that our coordinate system is
such that the y axis is perpendicular to the Earth’s
surface and points upward.
Mathematical Description
1
y(t) = − gt2 + v0y t + y0
2
vy (t) = −gt + v0y
Physics 2514 – p. 4/19
Clicker
An object is thrown vertically upward (assume the upward
direction is positive) with an initial velocity of 4.9 m/s. What is
the velocity and acceleration of the object when it reaches its
maximum height?
1. vy = 0 m/s
ay = 9.8 m/s2
2. vy = −4.9 m/s
ay = −9.8 m/s2
3. vy = 0 m/s
ay = 0 m/s2
4. vy = 0 m/s
ay = −9.8 m/s2
5. vy = −4.9 m/s
ay = 0 m/s2
Physics 2514 – p. 5/19
Clicker
A ball is thrown vertically upward (assume the upward direction
is positive) with an initial velocity of v0y . What is the velocity
when it returns to its starting point?
1. v0y
2. 2v0y
3. −v0y
4. 0 m/s
5. −2v0y
Physics 2514 – p. 6/19
Solution
There are two possible solution methods:
Use the equation for position verses time to determine the
total travel time, then use the equation for velocity verses
time to solve for the velocity. (These two equations
completely define the motion of the ball.)

1 2
y(t) = − gt +v0y t
2
⇒
0=
1
− gt + v0y t
2
taking y0 = 0, then
v(t) = −gt + v0y = −g
2v0y
g
⇒
t = 0
 t = 2v0y
g
+ v0y = −v0y
Physics 2514 – p. 7/19
Solution
Second method use the relation between velocity and
position
2
v 2 − v0y
= −2g(y − y0 )
⇒
v = ±v0y
where we take the starting position to be y0 = 0. Since the
final position is the starting position, then y = 0 also. The +
sign corresponds to its initial upward direction, while the −
sign corresponds to its final downward direction.
Physics 2514 – p. 8/19
What’s a Vector?
Vector Mathematical object with magnitude and direction;
Scalar Mathematical object that is represented by a single
number such as time, temperature, distance, . . .
A vector is defined independent of coordinate system but
most easily manipulated once a coordinate system is
defined
Some conventions
~
A vector is written as a variable with an arrow over it A
~ =A
The magnitude is written a script letter |A|
The magnitude of a component (to be defined) of a
vector is written as a script letter with a subscript Ax
Physics 2514 – p. 9/19
Vector Addition—Graphical
We previously discussed the graphical addition of vectors. The
idea is to place the tail of the vector on the tip of the preceding
vector in the sum, and the result is from tail to tip.
Physics 2514 – p. 10/19
Vector Subtraction—Graphical
Similar to addition, we can draw the difference of two vectors by
placing their tails at the same position and using the
parallelogram method.
eplacements
~
A
~
A
~
B
~
A
~
~ −B
A
~
−B
~
~ −B
A
~
B
~
−B
~
A
Physics 2514 – p. 11/19
Vector Addition—Algebraic
If two vectors are perpendicular to each other, the magnitude of
the sum is given by the Pythagorean theorem. The angle
relative to the first vector in the sum is found using the
trigonometric “tangent” function.
~ = PQ
The addition of the two vectors A
~ = QS is C
~ =A
~ +B
~
and B
The magnitude is given by
√
C = A2 + B 2 = 5 mi
The direction is given by
θ = tan−1
B
A
= 37◦
Physics 2514 – p. 12/19
Clicker
Two vectors of equal magnitude are perpendicular to each other.
Which of the following is true:
eplacements
~ + B|
~ > |A
~ − B|
~
|A
~
A
~
B
~ + B|
~ = |A
~ − B|
~
|A
~ + B|
~ < |A
~ − B|
~
|A
Physics 2514 – p. 13/19
Vector Addition—Algebraic
If two vectors are not perpendicular to each other, the
magnitude of the sum is given by the Law of Cosines. The angle
relative to the first vector in the sum is found also by using the
Law of Cosines.
The addition of the two vectors is
~ =A
~ +B
~
C
The magnitude is given by
p
C = A2 + B 2 − 2AB cos(ψ) = 147 m
~ and B
~
with ψ = 45◦ the angle between A
The direction is given by
p
B = A2 + C 2 − 2AC cos(φ)
h 2 2
i
A +C −B 2
−1
φ = cos
= 106◦
2AC
θ = 180◦ − φ = 74◦
Physics 2514 – p. 14/19
Vector Components
The standard method for solving problems is to reference
positions and directions off a coordinates system. (That is we
define a fixed position in space and a set of orthogonal
directions.) Let’s place a vector in a coordinates system.
~ is seen to be the sum of two
The vector A
~ x and A
~y
vectors A
~ x is the projection (shadow)
The vector A
on the x-axis.
It has no projection
(shadow) on the y-axis.
~ y is the projection (shadow)
The vector A
on the y-axis.
It has no projection
(shadow) on the x-axis.
Physics 2514 – p. 15/19
Vector Decomposition
Given the figure on the previous slide, and the earlier discussion
of the adding two orthogonal vectors, we can deduce a method
for decomposing a vector.
The components are defined in terms
of a right triangle. Using trigonometric
relations, we arrive at the components:
Opposite
Ay
=
Hypotenuse
A
Adjacent
Ax
cos θ =
=
Hypotenuse
A
sin θ =
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Vector Algebra
The purpose for decomposing: components in a given direction
can be added like ordinary numbers. Once components are
added, use Pythagorean theorem to calculate magnitude and
trig relations for direction.
Physics 2514 – p. 17/19
Vector Algebra
Let’s introduce unit vectors to simplify to simplify writing vector in
terms of components
The unit vectors have a magnitude equal
to 1, and a direction along one of the axis.
î parallel to x-axis
ĵ parallel to y-axis
Addition of two vectors:
~ = Ax î + Ay ĵ
A
~ = Bx î + By ĵ
B
~ =A
~ +B
~ = (Ax + Bx )î + (Ay + By )ĵ
C
Cx = A x + B x
Cy = A y + B y
Physics 2514 – p. 18/19
Assignment
Finish reading Chapter 3 and read section 2.7 motion on an
incline plane for the next lecture
Monday’s lecture will conclude the material in chapters 2 & 3.
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