Measurements & units Scalars & vectors Displacement, Velocity and acceleration Relative velocity.
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Transcript of Measurements & units Scalars & vectors Displacement, Velocity and acceleration Relative velocity.
1. Measurements & units2. Scalars & vectors3. Displacement, Velocity and
acceleration4. Relative velocity.5. Motion in two dimensions and
in three dimensions6. Special case: Gravity
Summary• Vectors: Positions, Displacement,
Velocity and Acceleration.
Vector or Scalar?Vector or Scalar?
• Speed………..
• Velocity……...
• Acceleration..
• Time………….
• Force…………
• Distance……..
scalar
vector
vector
scalar
scalar
it depends...
Some Derivatives
• Powers
• Trig Functions
• Exponentials
?xn dx
d
?)xsin( dx
d
?ex dx
d
Average Velocity What is the average velocity in the last
second (t = 3 to 4) ?
A. 2 m/sB. 4 m/sC. 1 m/sD. 0 m/s
x (meters)
t (seconds)
2
6
-2
4
1 2 43
Instantaneous velocity What is the instantaneous velocity in the
last second?
A.-2 m/sB. 4 m/sC. 1 m/sD. 0 m/s
x (meters)
t (seconds)
2
6
-2
4
1 2 43
Average Speed What is the average speed over the
first 4 seconds ?
A. 2 m/sB. 4 m/sC. 1,5 m/sD. 0 m/s
x (meters)
t (seconds)
2
6
-2
4
1 2 43
turning point
Correcting home exercises
What is displacement of a train from staring point to point at 3 seconds after ?
What is the velocity and acceleration of a train ?? (AV or IV) from staring point to point at 3 seconds after ?
Part 4
Relative velocity.
Air speed
Ground speed
Brainstorming
Galilean formula of velocity sum
Learning Check
Solution
a) Up
Learning Check
ground
river
boat
Part 5
Motion in one dimension and in two dimensions
Linear motion
0 is certain point
Green car with solar cell
Learning Check
John is moving to x direction by equation: X= - 25t2 +3t +7 (cm)
1- What is John ‘s position at time t=0? and t = 3(s) ? 2- What is his velocity at time time t=0? and t = 3(s) ? Average speed of John after 10s
moving? 3- What is his acceleration at time time t=0? and t = 3(s) ? Average acceleration of John after 10s
moving?
Learning checkAcceleration vs Time Plots
• Gives acceleration at any time.
• Area gives change in velocity Acceleration at t=4, a(4) =
Change of v between t=4 and t=1. v =
a
t4
3
-3
Constant Acceleration
Equation of motion isdt
dva
dvdt a
dtadv
tavv o
The o in v subscript refers to the original or initial value at the beginning of the time interval of interest.
Integrate both sides
where acceleration is constant.
Solution
Arranging this equationdt
dxv
dxdtv
dttavdx oo
dttavdx oo
2ooo ta
2
1tvxx
Substituting the velocity equation from the previous page
Integrating both sides
Learning Check
X=2+10t +4t2 (m)At t=3 x= 2+3.10 +4.9 = 68 (m)V= 10 + 8t (m/s)At x=4 4= 2+10t +4t2 4t2 +10t –2 =0 t = ?
Part 5
Motion in two dimensions
Motion in two dimensions
2y
2x
2 VVV 2y
2x
2 aaa
Positions in 2 dimensions
-100
-80
-60
-40
-20
0
20
0 5 10 15 20
v (m/s)
t (seconds)
-300
-200
-100
0
100
0 5 10 15 20
x (meters)
t (seconds)
• Where is velocity zero?• Where is velocity positive?• Where is velocity negative?• Where is speed largest?
• Where is acceleration zero?• Where is acceleration positive?
position vs. time
velocity vs. time
Example
Learning Check
Learning Check
Part 6
Free fall
Isaac Newton in 1689, by Sir Godfrey Kneller.
History
Learning Check
Exercises of today’s lecture
A ball is thrown straight up in the air and returns to its initial position. During the time the ball is in the air, which of the following statements is true?A - Both average acceleration and average velocity are zero.B - Average acceleration is zero but average velocity is not zero.C - Average velocity is zero but average acceleration is not zero.D - Neither average acceleration nor average velocity are zero.
Summary of Concepts
• kinematics: A description of motion• position: your coordinates• displacement: x = change of position• velocity: rate of change of position
– average : x/t– instantaneous: slope of x vs. t
• acceleration: rate of change of velocity– average: v/t– instantaneous: slope of v vs. t
Class Question
• How do units differ from variables?
List 10 clear examples of units and 10 clear examples of variables.
Problem
• A motorcycle moves with an initial velocity of 30m/s.
• When its brakes are applied, it decelerates at 5.0m/s2 until it stops.
• Plot the position, velocity and acceleration as a function of time.
• What is the position, velocity and acceleration 2 seconds after the brakes are applied?
2ooo ta
2
1tvxx
tavv o
Use bike computer
Bikebrain
Source: http://www.bikebrain.com
Attaches to a “PalmPilot”
Problem
• A car starts from rest and travels northward.• It accelerates at a constant rate for 30
seconds until it reaches a velocity of 55mph.• Plot the acceleration, velocity and position
as a function of time.
Problem
• A girl shoots an arrow upward.
• It strikes the ground 10.0 seconds later.
• What was its initial velocity and what was the maximum height?
2yoyoo ta
2
1tvyy
tavv yyoy
Problem
• A man standing on a 20-m helicopter throws a ball upward at 120 m/s.
• How long does it take to hit the ground?
Team Exercise, 3 min.
1. The derivative of velocity with respect to time is:– position or acceleration
2. By integrating velocity with respect to time we get:– distance traveled or acceleration
3. The derivative of position with respect to time is:– acceleration or velocity
4. Integrating acceleration twice with respect to time is :– velocity squared or distance
5. The derivative is associated with the _________ while the integral is associated with _________– area under the curve, slope
Team Exercise (3 minutes)
• One dimensional motion– What is the distance traveled in 3 seconds? – What is the acceleration at 1.25 hours?
Sp
eed
, mp
h
3210
010
20
Time, hours
Please
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