Download Thursday Aug 27 1-d Motion/Kinematics • Goal: Describe Motion

Thursday Aug 27
• Assign 1 – Due Friday 11:59 PM
• Lab next week – Print lab and do
pre-lab
• Help Room Th 6-9 PM
– Walter 245
• Office hours by appointment
– T.: 2-3 PM | Th.: 3-4 PM
• HW+Pre-Class = 20% of grade
1-d Motion/Kinematics
• Displacement
• Velocity
• Acceleration
• Goal: Describe Motion
• Kinematics – 'How'
• Dynamics – 'Why' (Forces)
• Supplemental instructor
Kevin Dempsey
• SI sessions in Morton 227
Monday 8:10-9:10 PM,
Tuesday 8:10-9:10 PM, and
Thursday 7:05-8:05 PM.
http://capa2.phy.ohiou.edu/res/ohiou/physlets/1dkinematics/four_opt.html
PhET
PhET
A runner runs 350 m due East, then turns around and runs 550 m
due West. What is the total displacement of the runner relative to
the starting position?
(1) 200 m West
(4) 350 m East
(7) 900 m West
200 m
(2) 200 m East
(5) 550 m West
(8) 900 m East
(3) 350 m West
(6) 550 m East
350m
550 m
Displacement is a vector – distance between two positions
What would be the total distance traveled?
900 m
Kinematics
• Goal: Describe Motion
• Kinematics – 'How'
• Dynamics – 'Why' (Forces)
• Position, Displacement, Velocity, Acceleration
– All vector quantities
– 3-d
• Distance traveled, speed
– scalar quantities
! Sect 2.1
Position, Displacement and Distance Traveled
! Sect 2.1
Need coordinate system:
• Axes
• Origin
Position:
Location relative to origin
Initial position (x0) = +1.5 m
Final position (xf) = +3.5 m
0
Origin
x=0, y=0
Displacement:
Change in Position
Δx = xf - x0
Displacement would be 2.0 m to the right
(Δ = change = final – initial)
(Capital greek Delta)
Distance Traveled
Ignore direction and add
up distance traveled
Displacement and Distance Traveled
! Sect 2.1
A runner runs 350 m due East, then turns around and runs 550 m
due West.
200 m
350 m
xf
x0
550 m
x=0m
Displacement:
Δx = xf - x0 = -200m – 0m = -200 m
(200 m to the left of the origin)
+x
Distance Traveled
350 m + 550 m = 900 m
Note: displacement not always (rarely) equals distance traveled
! Sect 2.2
Scalars and Vectors
Scalars
• Have magnitude
VECTORS
• Have magnitude and direction
Litmus test: does adding a direction make sense?
• A force of 10 N to the East ? fine – vector
• A mass of 10 kg to the West? nonsensical - scalar
Examples of Scalars:
Examples of Vectors:
• Time
• Force
• Mass
• Displacement
• Energy
• Velocity
Scalars and Vectors
! Sect 2.2
Why do we care?
If vectors, must be more careful mathematically.
Scalars add, subtract, multiply the way we're used to.
Vectors need to account for direction – may need trig.
If parallel or antiparallel, then can treat more simply.
If not – need to use trig.
Magnitude of vector is its ‘size’ (ignore direction)
The magnitude of a vector “3 m to the left” is 3 m
Book will represent vectors as bold face (x, v, a)
Vector magnitudes will be italicized (x, v, a)
! Sect 2.1
Velocity, Speed
• Velocity: how fast and in what direction (vector)
• Speed: how fast (scalar)
• Ave. Velocity = Displacement/Elapsed time
– vector
– only depends on initial and final position
– same direction as displacement
– bar over v represents average
Δx
v=
Δt
x f − x0
v=
t f − t0
x f = x0 + vΔt
• Ave. Speed = Distance traveled / Elapsed time
– no mention of direction
– Ave. Speed is not the magnitude of average velocity
Motion: “Multiple Representations”
We can represent a motion in a variety of ways:
• Text: “A squirrel runs at a constant speed of 2 m/s in a straight line.”
• Math: xf = x0 + vΔt = x0 + (2m/s) Δt
• Animation
• Motion Diagram: (snapshots at even time intervals)
position
velocity
• Graph:
time
time
Graphing Basics: Slope
y
• Slope = rise/run = Δy / Δx
• Up to right is positive
• Down to right is negative
• Slope of curve at a point
– slope of tangent line
(tangent line to a plane
curve at a given point is y
the straight line that
"just touches" the curve
at that point.)
• Slope of straight line same
at any point
Phet
rise = Δy
run = Δx
x
A
x
The slope at point B on this curve is _________ as you move to the
right on the graph. Think about slope just a 'smidgen' to the left and
right of Point B.
y
1. increasing
B
2. decreasing
3. staying the same
x
Slope of a straight line is the same anywhere along that line.
Along the straight portion of the curve, the slope is not changing.
(1) A
(2) D
(3) C
(4) A and D
(5) B and E
(6) A, B, D and E
position
This position versus time graph represents the position of a
person as a function of time. At what point or points is the
person standing still?
C
A
D
E
B
time
At A and D the position is not changing
Moving Man PhET
! Sect 2.2
Average vs. Instantaneous
• Average – need beginning and end 'events' (times)
– 'event' – time, place, velocity, accel
• Instantaneous – that instant
– shrink Δt to very small
• Slope on position vs. time
graph is change in position
divided by change in time
PhET
Δx
v = lim Δt→0
Δt
The plot represents the position of a runner as a function of time.
What is the average velocity of the runner during the time interval
from 1 to 4 secs?
(1) 5 m/s
(2) 6.7 m/s
(3) 7.5 m/s
(4) 10 m/s
Δx x f − x0 30m−10m
v=
=
=
= 6.7m/ s
Δt t f − t0
4s−1s
What is the velocity of the runner in t=0, t=1 and t=2 sec?
(1) A
(2) D
(3) C
(4) A and D
(5) B and E
(6) A, B, D and E
position
This position versus time graph represents the position of a
person as a function of time. At what point or points is the
velocity constant (not changing)?
C
A
D
E
B
time
Slope represents the velocity. Anywhere slope is not
changing, velocity is not changing!
At what point is the person moving fastest?
B – greatest slope
Which velocity plot corresponds to
the position plot to the right?
! Sect 2.7
x vs t graph: Constant positive slope, then zero slope (zero v)
At Point B the velocity is:
zero
constant
increasing
decreasing
D
position
(1)
(2)
(3)
(4)
E
C
A
B
time
Slope represents the velocity.
At Point B the slope is increasing as time increases.
! Sect 2.3
Acceleration
• change in velocity over change in time
• Vector
Δv
a=
Δt
v f = v0 + aΔt
Δv
a = lim Δt→0
Δt
• Same direction as CHANGE in velocity
• acceleration related to Force (a=F/m)
• If net force constant, acceleration is constant
v
t
Note: acceleration would be slope on v vs. t graph (changing velocity)
Would be change in slope on x vs. t graph
Think about how velocity is changing
2 objects are accelerating over a time period of 6 seconds. A toy rocket changes its velocity from 140 m/s to 150 m/s. A bicyclist changes its velocity from 5 m/s to 15 m/s. Which has the greatest acceleration?
1. The bicyclist has the greater acceleration.
2. The rocket has the greater acceleration.
3. They both have the same acceleration.
Same Δv, same Δt
Δv 150m/s − 140m/s
a=
=
= 1.67 m/s 2
Δt
6s
Units on LON-CAPA: m/s^2
Uniform Motion
• constant velocity
• constant speed in constant direction
• Motion diagram of two balls (both are moving to the right)
A
uniform
B
non-uniform
Which ball is undergoing non-uniform motion?
1-D Kinematics
• Goal:
– Understand direction (and in 1-d, the sign) of acceleration
– Try to represent the motion graphically
•
•
http://capa2.phy.ohiou.edu/res/ohiou/physlets/1dkinematics/four_opt.html
http://capa2.phy.ohiou.edu/res/ohiou/physlets/1dkinematics/accel_prs.html
If acceleration and velocity in same direction, speeding up.
If acceleration and velocity in opposite directions, slowing down.
Speed is magnitude of velocity (no direction)
! Sect 2.4
• Const net force → Const accel
Δx = vΔt
Δv = aΔt
If a = const, a = a
Simplification:
ti = 0 restart clock
i→0 initial becomes t=0
x, v, a at time t (final event)
x = x0 + vt
1
v = (v0 + v)
2
v = v0 + at
1 2
x = x0 + v0 t + at
2
v2 = v02 + 2a (x − x0 )
ONLY IF ACCEL IS CONST
Problem Solving – Basic Outline
•
•
•
•
•
•
•
! Sect 2.5
Read – Read
Draw/Imagine – lay out coordinate system
What know? Don't know? Want to know?
Physical Processes? Laws? Conditions?
Valid Relationships/Formulas
x = x0 + vt
Solve
1
v = (v0 + v)
Is it reasonable?
2
v = v0 + at
1 2
x = x0 + v0 t + at
2
v2 = v02 + 2a (x − x0 )
ONLY IF ACCEL IS CONST
! Sect 2.5
Example: Plane Landing
A jet plane lands with a velocity of +100 m/s and with full brakes on
can accelerate at a rate of -5.0 m/s2.
a) From the instant it touches the runway, what is the minimum time
needed before it can come to rest?
b) Can this plane land on a small island airport where the runway is
0.80 km long?
x=?
v0 = ?
initial?
final?
v = v0 + at
0 = (+100m/s) + (−5.0m/s)t
t = 20s
v =?
a =?
v2 = v02 + 2a(x − x0 )
t =?
(0m/s)2 = (+100m/s)2 + 2(−5.0m/s2 )x
x = 1000m
! Sect 2.5
Example:
A car accelerates uniformly from rest to a speed of 25 m/s in 8.0 s.
Find the distance the car travels in this time and the constant acceleration of the car.
Define the +x direction to be in the direction of motion of the car.
x=?
v0 = ?
x=
1
(v0 + v )t x = 100 m
2
v =?
a =?
t =?
v = v0 + at a = 3.125 m/s2
v = v0 + at
x=
1
(v0 + v )t
2
1
x = v0t + at 2
2
v 2 = v02 + 2ax
Which velocity plot corresponds
to the acceleration plot to the
right?
Accel constant, slope of v vs. t must be constant
Accel >0, slope must be greater than zero
! Sect 2.7
Which displacement plot corresponds to
the acceleration plot to the right?
Velocity increases – slope of x vs. t should increase
When x vs t function curves upward, positive acceleration.
When x vs t function curves downward, negative acceleration
If curves over shorter time, greater acceleration!
Graphs - Summary
• Be careful – check axes
• Slope of pos vs time gives velocity
• If slope of pos vs time changing, velocity
changing, therefore accel non-zero.
(curvature)
• Slope of velocity vs time gives
acceleration
• (Area under v vs. t gives displacement)
PhET