Physics 2514
Lecture 22
P. Gutierrez
Department of Physics & Astronomy
University of Oklahoma
Physics 2514 – p. 1/15
Information
Information needed for the exam
Exam will be in the same format as the practice with the
same number of questions
Bring a # 2 pencil & eraser
Calculators will be allowed
No cell phones, no laptops, . . .
Only exam, pencil, eraser, calculator allowed on desk.
Bring student id with you
You will need to know
Student id number
Discussion section #
Your name
Physics 2514 – p. 2/15
Exam 2
Exam 2 will cover material in lectures 11 through 21. This
includes material in Chapters 5 through 8
Kinematics
Projectile motion & general 2-d motion with constant
acceleration;
Circular motion;
Relative motion.
Dynamics
Newton’s 3 laws of motion.
Physics 2514 – p. 3/15
Formulas to be given
The following formulas will be given
1 2
s(t) = at + v0 t + s0
2
v(t) = at + v0
v 2 − v02 = 2a(s − s0 )
s = rθ
~
~v = ~v0 + V
~ net = m~a
F
fk = µn
fs ≤ µn
Physics 2514 – p. 4/15
Review Kinematics
Started with discussion of motion
Displacement final minus initial position, with direction
pointing from initial to final position
~r = ∆~r = ~rf − ~ri
Velocity is the rate of change in position
~vavg =
∆~r
∆t
⇒
~v(t) =
d~r
dt
Acceleration is rate at which the velocity changes
~aavg =
∆~v
∆t
⇒
~a(t) =
d~v
dt
Physics 2514 – p. 5/15
Review 2-d Kinematics
Kinematic equations for constant acceleration (Apply
independently in each dimension)
Position
s(t) =
1 2
at + v0 t + s0
2
Velocity
v(t) = at + v0
Combining equations
v 2 − v02 = 2a(s − s0 )
Find constraint that ties equations together
How long does it take to fall, how far does it travel horizontally
Physics 2514 – p. 6/15
Steps in Problem Solving
Steps in problem solving
A) Rewrite the problem eliminating all extraneous
information. (What are you given, what are you looking,
what are the constraints);
B) Draw a diagram along with a coordinate system, label
each object with the variables associated with it;
C) What are the known and unknown quantities, which
unknowns are you solving for;
D) Write down the equations associated with the problem,
and solve the problem algebraically (SIMPLIFY!!!)
E) Finally, substitute numbers into the equation, and
calculate the numerical solution
Physics 2514 – p. 7/15
Example
A catapult is tested by Roman legionnaires. They tabulate the
results in a papyrus scroll and 2000 years later an
archaeological team reads (distance translated into modern
units): Range = 0.20 km; angle of launch = π/3. What is the
initial velocity of launch of the boulder?
Object launched at angle π/3 radians, travels 0.2 km. Determine
initial velocity using the constraint, how long does it take to hit
the ground?
PSfrag replacements
v0
θ
vy
v0
θ
vx
Physics 2514 – p. 8/15
Example
Object launched at angle π/3 radians, travels 0.2 km. Determine initial
velocity using the constraint, how long it take to hit the ground?
PSfrag replacements
v0
θ
v0
vy
θ
vx

1 2
yf = − gtf + (v0 sin θ)tf 
2

xf = (v0 cos θ)tf
⇒
s


2xf tan θi − 2yf


= 8.4 s

 tf =
g
p


xf gxf tan θi − gyf


√
v
=
= 47.6 m/s
 0
2(xf sin θ − yf cos θ)
xf ≡ x(tf ) = 200 m, yf ≡ y(tf ) = 0 m,
θi ≡ θ(ti ) = π/3 rads
Physics 2514 – p. 9/15
Circular Motion
Tangential & Angular Velocity
Arc-Length & Angle
s = rθ
v=
Uniform Motion |~
v|
dθ
ds
= rω
=r
dt
dt
= constant
s(t) = v0 t + s0
⇒
θ = ω0 t + θ0
v(t) = v0
⇒
ω(t) = ω0
Physics 2514 – p. 10/15
Nonuniform Circular Motion
Consider that case when the speed around a circle changes.
Physics 2514 – p. 11/15
Kinematic Equations
The kinematic equations for uniform circular motion were
derived earlier, here we consider nonuniform motion
Motion along arc is 1-D with tangential
acceleration and velocity determining motion
s = s0 + vot t +
1
at t 2
2
vt = v0t + at t
Now divide by r
1
1
(s = s0 + v0t t + at t2 )
r
2
1
(vt = v0t + at t)
r
θ = θ 0 + ω0 t +
ω = ω0 + αt
1 2
αt
2
α = at /r
Physics 2514 – p. 12/15
Centripetal Acceleration
Calculate average acceleration
CB = ∆~r2 −∆~r1 = ~
v2 ∆t−~
v1 ∆t = ∆~
v∆t
Angles
ABO: θ + α + α = 180
DAC: φ + α + α = 180
)
⇒
θ=φ
Similar triangles
AB
CB
=
AB
AO
⇒
|∆~
v|∆t
v∆t
=
v∆t
r
Average radial acceleration
average
ar
|∆~
v|
v2
=
=
∆t
r
|∆~
v|
v2
ar = lim
=
∆t→0 ∆t
r
Physics 2514 – p. 13/15
Example
A car starts from rest on a curve with a radius of 120 m and
accelerates at 1.0 m/s2 . Through what angle will the car have traveled
when the magnitude of its total acceleration is 2.0 m/s2 .
replacements
Knowns
vt
at
ar
at = 1.0 m/s2
p
af = a2t + a2r = 2.0 m/s2
vt0 = 0 m/s
r = 120 m
θ0 = 0 rads
Unknowns
θf = ? rads
Physics 2514 – p. 14/15
Example
A car starts from rest on a curve with a radius of 120 m and
accelerates at 1.0 m/s2 . Through what angle will the car have traveled
when the magnitude of its total acceleration is 2.0 m/s2 .
Knowns
at = 1.0 m/s2
p
af = a2t + a2r = 2.0 m/s2
vt0 = 0 m/s
r = 120 m
θ0 = 0 rads
Unknowns
θf = ? rads
How long to reach af
r
v 2 2
2
af = at + rtf
= 2.0 m/s2
vtf = 14.4 m/s
vtf = at tf ⇒ tf = 14.4 s
How far
θf = 12 αt2f =
1 at 2
2 r tf
⇒ θf = 0.864 rads
Physics 2514 – p. 15/15