Transcript Classical Mechanics - United States Naval Academy

Classical Mechanics
• Kinematics – how
objects move
– Translational motion
– Rotational motion
– Vibrational motion
• Dynamics – Forces
and why objects move
as they do
– Statics – special case forces cause no motion
Reference Frames
• Position, distance or speed must be
specified wrt some reference frame.
• Coordinate axis
• Position = coordinate
• Displacement = change in position
– final minus initial
• Displacement is a vector! Direction
is from the initial to final position.
x  x 2  x1
Average Velocity
x x 2  x1
v

t
t 2  t1
More Examples
x x 2  x1
v

t
t 2  t1
x dx
v  lim

t 0 t
dt
Velocity
• Average velocity
x x 2  x1
v

t
t 2  t1
• Instantaneous velocity
x dx
v  lim

t 0 t
dt
L
 T 
Velocity vs. Speed?
Instantaneous Velocity
x dx
v  lim

t 0 t
dt
Acceleration
• Average
acceleration
v v2  v1
a

t t 2  t1
• Instantaneous
acceleration
v dv
a  lim

t 0 t
dt
L
 T 2 
dv d  dx  d 2 x
a
   2
dt dt  dt  dt
Example Problem 1
• A jogger runs in a straight line with
– an average velocity of 5.00 m/s for 4.00 min.
– an average velocity of 4.00 m/s for 3.00 min.
• What is her total displacement?
• What is her average velocity during this
time?
Example Problem 2
• A particle moves along the x-axis acording to the equation (x
is in meters, and t is in seconds):
2
x  2t  3t
•
•
•
•
•
What must be the units on the constants?
What is the instantaneous velocity at t = 3.0 s?
What is the instantaneous acceleration at t = 3.0 s?
What is the displacement between t = 2.0 s and t = 3.0 s?
What is the average velocity between t = 2.0 s and t = 3.0 s?
Motion Diagrams
See Wiley Plus Concept Simulation
1D constant acceleration
Difference between constant acceleration
and constant velocity
What if acceleration is known?
t
v  v0   adt
0
t
x  x 0   vdt
0
Memory Aid
d2x
a 2
dt
dx
dt
dv
dt
x, x
v, v
a
 vdt
 adt
Constant Acceleration
v  v0  at
1 2
x  x 0  v 0 t  at
2
v  v  2a  x  x0 
2
2
0
v  v0
v
2
Example Problem 3
• A car will go 0 to 87 mph in 8 sec. Assume
the acceleration is constant.
• Find the acceleration
• Find the position after 8 sec.
• Find the velocity after 10 sec.
Falling Objects
• Galileo: at a given location on the earth and
in the absence of air resistance, all objects
fall with the same constant acceleration.
m
g     9.8 2
s
Example Problem 4
• A person throws a ball upward into the air
with an initial velocity of 15.0 m/s.
• Find:
– How high it goes
– How long the ball is in the air before it comes
back to his hand
– How much time it takes to reach maximum height
– The velocity of the ball when it returns to the
throwers hand
– At what time the ball passes a point 8.00 m above
the person’s hand.
The Graphs
Deceleration
• My golf ball goes flying in to the sandtrap
at 30.0 m/s. It looks like a fried egg buried
3.00 cm into the sand. What was the
acceleration of the golf ball caused by the
sand as it landed?
Homework
• Wiley Plus Chap 2