The document discusses finding velocity and acceleration from position equations in physics, covering concepts such as average velocity and constant acceleration. It explores relative motion through examples of different reference frames and transformations of position and velocity. Notable distinctions between Galileo's and Einstein's theories on velocity transformation are highlighted, emphasizing that acceleration remains consistent across inertial frames.

1. Finding v(t) and a(t) from r(t)
Suppose we have an equation in powers of t that
gives the position r of an object vs. time. How can we
find the velocity v and acceleration a?
2 2
ˆ ˆ ˆ ˆ ˆ ˆ( ) x y x y x yr t a x a y b tx b ty c t x c t y= + + + + + +
r
L
2 2
ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( )x y x y x yr t t a x a y b t t x b t t y c t t x c t t y+ ∆ = + + + ∆ + + ∆ + + ∆ + + ∆
r
2 2
ˆ ˆ ˆ ˆ( , ) ( ) ( ) (2 ) (2 )x y x yr t t r t t r t b tx b ty c t t t x c t t t y∆ ∆ = + ∆ − = ∆ + ∆ + ∆ + ∆ + ∆ + ∆
r r r
0
ˆ ˆ ˆ ˆ( ) lim 2 2x y x y
t
r
v t b x b y c tx c t y
t∆ →
∆
= = + + +
∆
r
r
ˆ ˆ( , ) 2 2x yv t t c tx c t y∆ ∆ = ∆ + ∆
r
0
ˆ ˆ( ) lim 2 2 constantx y
t
v
a t c x c y
t∆ →
∆
= = + =
∆
r
r
January 13, 2012 1/20Physics 114A - Lecture 7

2. Vector Motion with
Constant Acceleration
Average velocity:
1
02
( )avv v v= +
r r r
1
0 0 02
( ) ( )avr t r v t r v v t= + = + +
r r r r r r
1 2
0 0 2
( )r t r v t at= + +
r r r r
Position as a function of time:
Velocity as a function of time:
0( )v t v at= +
r r r
January 13, 2012 2/20Physics 114A - Lecture 7

3. Graphs of x-y and v-t
t (s) x (m) y (m) v (m/s)
0 0 5 5.0
1 2 10 6.4
2 8 15 9.4
3 18 20 13.0
4 32 25 16.8
2
2 2
2 ; 4 /
(5 5) ; 5 /
x
y
x y
x t m v t m s
y t m v m s
v v v
= =
= + =
= +
January 13, 2012 3/20Physics 114A - Lecture 7

4. Clicker Question 1
Which of the blue position vs.
time graphs goes with this green
velocity vs. time graph? The
particle’s position at ti = 0 s is
xi = −10 m.
January 13, 2012 4/20Physics 114A - Lecture 7

5. Relative Motion
Definition: An Inertial Reference Frame is any coordinate system
(or frame of reference) that is either at rest or moving in some
direction with a constant velocity.
Example: a train or airplane moving forward with a constant speed.
January 13, 2012 5/20Physics 114A - Lecture 7

6. Relative Motion
The speed of the passenger with respect to
the ground depends on the relative directions
of the passenger’s and train’s speeds:
vgnd = 16.2 m/s vgnd = 13.8 m/s
January 13, 2012 6/20Physics 114A - Lecture 7

7. Relative Motion
This also works in two dimensions:
January 13, 2012 7/20Physics 114A - Lecture 7

8. Relative Motion
Amy, Bill, and Carlos all measure the
velocity of the runner and the acceleration
of the jet plane. The green velocity vectors
are shown in Amy’s reference frame.
What is the runner’s velocity? It
depends on the frame of the observer.
Amy: vR = 5 m/s
Bill: vR = 0 m/s
Carlos: vR =−10 m/s
What about aplane?
January 13, 2012 8/20Physics 114A - Lecture 7

9. Relative Position
' ' ;
' ;
r r R r Vt
r r R r Vt
= + = +
= − = −
r rr r r
r rr r r
' ; ' ;
' ; ' ;
x x
y y
x x V t x x V t
y y V t y y V t
= + = −
= + = −
The position depends on the frame. A set of
position observations in one frame can be
transformed to get equivalent observations in
another frame.
January 13, 2012 9/20Physics 114A - Lecture 7

10. Relative Velocity
pB pA ABv v v= +
r r r
pG pC CGv v v= +
r r r
A
z
x
y
B y’
z’
x’
vAB
vpA
The velocity also depends on the frame. A set of velocity
observations in one frame can be transformed to get equivalent
observations in another frame.
January 13, 2012 10/20Physics 114A - Lecture 7

11. Example: Flying a Plane
pG pA AGv v v= +
r r r
A pilot wishes to fly a plane due north relative to the
ground. The airspeed of the plane is 200 km/h, and
the wind is blowing from west to east at 90 km/h.
(a) In which direction should the plane head?
(b) What will be the ground speed of the plane?
(90 km/h)
arcsin arcsin 26.7 west of north
(200 km/h)
AG
pA
v
v
θ = = = °
2 2 2 2
(200 km/h) (90 km/h) 179 km/hpG pA AGv v v= − = − =
January 13, 2012 11/20Physics 114A - Lecture 7

12. Example: Crossing a River
You are riding in a boat with a speed
relative to the water of vbw = 6.1 m/s. The
boat points at an angle of θ = 25° upstream
on a river flowing at vwg = 1.4 m/s.
(a) What is your speed vbg and angle θbg
relative to the ground?
bg bw wgv v v= +
r r r
ˆ( 1.4 m/s)wgv y= −
r
ˆ ˆ(6.1 m/s)cos25 (6.1 m/s)sin 25
ˆ ˆ(5.5 m/s) (2.6 m/s)
bwv x y
x y
= ° + °
= +
r
ˆ ˆ(5.5 m/s) (2.6 m/s 1.4 m/s)
ˆ ˆ(5.5 m/s) (1.2 m/s)
bgv x y
x y
= + −
= +
r
2 2
(5.5 m/s) (1.2 m/s) 5.6 m/sbgv = + =
[ ]1
tan (1.2 m/s) / (5.5 m/s) 12bgθ −
= = °
January 13, 2012 12/20Physics 114A - Lecture 7

13. Example: A Ball Toss
Mike throws a ball upward at a 630
angle with a speed of 22 m/s. Nancy
rides past Mike on her bicycle at 10 m/s
at the instant he releases the ball.
(a) What trajectory does Mike see?
(b) What trajectory does Nancy see?
0 0
0 0
0 0 0
1 2 2
0 0 0 02
max
Mike:
cos (22 m/s)cos 63 =10.0 m/s;
sin (22 m/s)sin 63 =19.6 m/s;
( ) 10.0 m;
( ) ( ) (19.6 - 4.9 ) m;
=19.6 m at 2 s and 0 at 4 s.
x
y
x
y
v v
v v
x x v t t t
y y v t t g t t t t
y t y t
θ
θ
= = °
= = °
= + − =
= + − − − =
= → =
Nancy:
' (10.0 -10.0 )m = 0 m;
' ;
Therefore, the ball rises and falls vertically
and Mike moves backward at 10 m/s.
x
y
x x V t t t
y y V t y
= − =
= − =
January 13, 2012 13/20Physics 114A - Lecture 7

14. The Object and Frame
Velocities Add Vectorially
January 13, 2012 14/20Physics 114A - Lecture 7

15. Consider how the acceleration transforms from frame S to frame S’,
an inertial frame that is moving with constant velocity V relative to S. An
“inertial frame” is defined to be any reference frame that is at rest or
moving with a constant velocity.
Velocities add, so
Galilean Relativity
'
' ; 0;
'
'
'
v v V v
v v V
t t t t
v v
a a
t t
a a
∆ ∆ ∆ ∆
= + = + = +
∆ ∆ ∆ ∆
∆ ∆
= = =
∆ ∆
=
rr r rrr r
r r
r r
r r
Galilean Relativity: While position and velocity are frame-
dependent, acceleration is observed to be the same in all inertial
reference frames.
January 13, 2012 15/20Physics 114A - Lecture 7

16. Galileo vs. Einstein
The laser beam moves along
the x axis away from Tom at the
speed of light, vx = 3 x 108
m/s.
Sue flies by in her space ship,
moving along the x axis at Vx =
2 x 108
m/s. From her point of
view, how fast is the laser beam
moving?
Galileo: vx’ = vx – Vx = 1 x 108
m/s
Einstein: vx’ = vx = 3 x 108
m/s
Velocity transformations are valid
in all inertial reference frames.
The speed of light is the same in
all inertial reference frames.
January 13, 2012 16/20Physics 114A - Lecture 7

17.  Before the next lecture on Tuesday, read
Walker, Chapter 4.1 and 4.2
 Homework Assignments #1 is now due at
11:59 PM on Tuesday, January 17. Homework
Assignments #2 is still due at 11:59 PM on
Thursday, January 19.
 No class on Monday due to the MLK
Holiday. We will have Exam 1 on Friday,
January 20. Send seat requests, if you have
not already done so.
End of Lecture 7