# General Physics I Vectors & 2D Motion

```General Physics I
Vectors & 2D Motion
Vectors
In 1 dimension, we could specify direction with a + or - sign.
In 2 (or 3) dimensions, we need more than a sign to specify the
direction of something:
To illustrate this, consider the position vector r in 2 dimensions.
Example Where is Prestonsburg?
Example:
Choose origin at Pikeville
Choose coordinates of
distance (miles), and
direction (N,S,E,W)
In this case r is a vector that
points 25 miles north.
Page 1
Prestonsburg
r
Pikeville
Vectors...
There are two common ways of indicating that something is
a vector quantity:
Boldface notation: A
A= A
“Arrow” notation: A
Vectors...
The components of r are its (x,y) coordinates
r = (rx ,ry ) = (x,y)
We will consider this in 2-D:
rx = x = r cos θ
where r = |rr |
ry = y = r sin θ
(x,y)
y
θ = arctan (y / x)
= tan-1 (y / x)
r
θ
x
Page 2
Vectors...
The magnitude (length) of r is found using the Pythagorean
theorem:
r
y
r = r = x2 + y 2
x
The length of a vector clearly does not depend on its
direction.
Consider the vectors A and B. Find A + B.
A
A
B
A
B
C=A+B
B
We can arrange the vectors as we want, as long as we
maintain their length and direction!!
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Consider C = A + B.
Comparing components of (a) and (b):
Cx = Ax + Bx
Cy = Ay + By
By
C
B
A
θ
Ax
Page 4
Bx
Ay
For Triangle A:
We would find a similar solution for Triangle B.
Therefore:
Cx = Ax + Bx = A cos q + B cos q
Cy = Ay + By = A sin q + B sin q
C = Cx 2 + Cy 2
&
q=
tan-1 (Cy / Cx)
For Example...
A = 10 m, B = 20 m, and C = 7 m
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Solution...
2D Kinematics
We will deal with 2-D problems when acceleration is
constant:
Choose y axis to be along direction of acceleration
Choose x axis to be along the “other” direction of
motion
Example Throwing a baseball (neglecting air resistance)
Example:
Acceleration is constant (gravity)
Choose y axis up: ay = -g
Choose x axis along the ground in the direction of the
throw
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“x” and “y” components of motion are
independent.
A man on a train tosses a ball straight up in the air.
View this from two reference frames:
Reference frame
on the moving train.
Reference frame
on the ground.
Velocity Components
Page 7
The “x” & “y” Velocity Vectors
Page 8
Problem: Fall Time
Recall
that
there is
no
“gravity”
in the xdirection
!
So…Which
kid hits the
water first?
Problem: Sammy’s Homer??
Sammy Sosa clobbers a fastball toward center-field. The
ball is hit 1 m (yo ) above the plate, and its initial velocity is
36.5 m/s (vo) at an angle of 30o (θ
θ) above horizontal. The
center-field wall is 113 m (D) from the plate and is 3 m (h)
high.
What time does the ball reach the fence?
Does Sammy get a home run?
v
y0
h
θ
D
Page 9
Solution...
Shooting the Monkey
(tranquilizer gun)
Where does the zookeeper
aim if he wants to hit the monkey?
( He knows the monkey will
let go as soon as he shoots ! )
Page 10
Shooting the Monkey...
If there were no gravity, simply aim
at the monkey
End of
Vectors & 2D Motion
Lecture
Page 11
```