Department of Physics and Applied Physics95.141, F2010, Lecture 3
Physics I95.141
LECTURE 39/13/10
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Exam Prep Question• 2 cars are racing. Car A begins accelerating (aA=4m/s2), but
Car B stalls. After 5s, Car A thinks it has won and stops accelerating, while Car B finally starts and accelerates at a rate of aB=5m/s2.
(a) (5pts) What is the speed of Car A when Car B finally starts moving?
(b) (5pts) What is the head start (in m) that Car A gets?
(c)(10 pts) How long (in s) until Car B catches up to Car B?
(d)(10 pts) What is the minimum length of the race track required for Car B to win the race?
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Exam Prep Question
2 cars are racing. Car A begins accelerating (aA=4m/s2), but Car B stalls. After 5s, Car A thinks it has won and stops accelerating, while Car B finally starts and accelerates at a rate of aB=5m/s2.
– Draw Diagram/Coord. System– Knowns and unknowns
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Exam Prep Question
2 cars are racing. Car A begins accelerating (aA=4m/s2), but Car B stalls. After 5s, Car A thinks it has won and stops accelerating, while Car B finally starts and accelerates at a rate of aB=5m/s2.
(a) (5pts) What is the speed of Car A when Car B finally starts moving?(b) (5pts) What is the head start (in m) that Car A gets?
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Exam Prep Question
2 cars are racing. Car A begins accelerating (aA=4m/s2), but Car B stalls. After 5s, Car A thinks it has won and stops accelerating, while Car B finally starts and accelerates at a rate of aB=5m/s2.
(c) (10 pts) How long (in s) until Car B catches up to Car B?
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Exam Prep Question
2 cars are racing. Car A begins accelerating (aA=4m/s2), but Car B stalls. After 5s, Car A thinks it has won and stops accelerating, while Car B finally starts and accelerates at a rate of aB=5m/s2.
(d) (10 pts) What is the minimum length of the race track required for Car B to win the race?
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Outline
• Freely Falling Body Problems• Vectors and Scalars• Addition of vectors (Graphical)• Adding Vectors by Components• Unit Vectors
• What Do We Know?– Units/Measurement/Estimation– Displacement/Distance– Velocity (avg. & inst.), speed– Acceleration
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Review of Lecture 2
• Last Lecture (2) we discussed how to describe the position and motion of an object
• Reference Frames• Position• Velocity• Acceleration• Constant Acceleration
2
)(2
2
1
22
2
o
oo
oo
o
vvv
xxavv
attvxx
atvv
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Freely falling Bodies
• Most common example of constant acceleration is a freely falling body.
• The acceleration due to gravity at the Earth’s surface is basically constant and the same for ALL OBJECTS (Galileo Galilei)
28.9s
mgag
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Example Problem• Batman launches his grappling bat-hook upwards, if the beam it
attaches to is 50m above Batman’s Batbelt, at what bat-velocity must the hook be launched at in order to make it to the beam? (Ignore the mass of the cord and air resisitance)
1) Choose coordinate system2) Knowns and unknowns3) Choose equation(s)
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Example Problem• Batman launches his grappling bat-hook upwards, if the beam it
attaches to is 50m above Batman’s Batbelt, at what bat-velocity must the hook be launched at in order to make it to the beam? (Ignore the mass of the cord and air resisitance)
3) Choose equation(s)
4) Solve
2
)(2
2
1
22
2
o
oo
oo
o
vvv
xxavv
attvxx
atvv
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Vectors and Scalars
• A quantity that has both direction and magnitude, is known as a vector.– Velocity, acceleration, displacement, Force,
momentum– In text, we represent vector quantities as
• Quantities with no direction associated with them are known as scalars– Speed, temperature, mass, time
r,a,v
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Vectors and Scalars
• In the previous chapter we dealt with motion in a straight line– For horizontal motion (+/- x)– For vertical motion (+/- y)
• Velocity, displacement, acceleration were still vectors, but direction was indicated by the sign (+/-).
• We will first understand how to work with vectors graphically
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Vectors
• Graphically, we can depict a vector as an arrow– Arrows have both length (magnitude) and direction.
-5 5
-5
5y-axis
x-axis
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Addition of vectors
• In one dimension– If the vectors are in the same direction
– But if the vectors are in the opposite direction
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Addition of Vectors (2D)
• In two dimensions, things are more complicated
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Addition of Vectors
• “Tip to tail” method– Draw first vector– Draw second vector, placing tail at tip of first vector– Arrow from tail of 1st vector to tip of 2nd vector is
)(2
)(3
2
1
ymD
xmD
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Commutative property of vectors
• “Tip to tail” method works in either order– Draw first vector– Draw second vector, placing tail at tip of first vector– Arrow from tail of 1st vector to tip of 2nd vector is
)(2
)(3
2
1
ymD
xmD
1221 DDDD
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Three or more vectors
• Can use “tip to tail” for more than 2 vectors
+ + =
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Subtraction of vectors
• For a given vector the negative of the vector is a vector with the same magnitude in the opposite direction.
1V
1V
)( 2121 VVVV
- = +
• Difference between two vectors is equal to the sum of the first vector and the negative of the second vector
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Adding vectors by components
• Adding vectors graphically is useful to understand the concept of vectors, but it is inherently slow (not to mention next to impossible in 3D!!)
• Any 2D vector can be decomposed into components
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Determining vector components
• So in 2D, we can always write any vector as the sum of a vector in the x-direction, and one in the y-direction.
• Given V(V,θ), we can find Vx and Vy
yx VVV
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Determining vector components
• Or, given Vx and Vy, we can find V(V,θ).
x
y
yx
V
V
VVV
tan
22
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Example
• A vector is given by its vector components:
• Write the vector in terms of magnitude and direction
4,2 yx VV
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Adding vectors by components
• Given V1 and V2, how can we find V= V1 + V2?
yx
yyxx
VV
VVVV
VVV
2121
21
V1
V2V
Department of Physics and Applied Physics95.141, F2010, Lecture 3
3D Vectors
• Adding vectors vectors by components is especially helpful for 3D vectors.
• Also, much easier for subtraction
zzyyxx
zyxzyx
VVVVVVVVV
VVVVVVVV
21212121
22221111 ,
zzyyxx
zyxzyx
VVVVVVVVV
VVVVVVVV
21212121
22221111 ,
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Multiplying a vector by a scalar
• You can also multiply a vector by a scalar
• When you do this, you don’t change the direction of the vector, only its magnitude
1Vc
c=2 c=4 c=-2
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Unit Vectors
• Up to this point, we have written vectors in terms of their components as follows:
• There is an easier way to do this, and this is how we will write vectors for the remainder of the course:
vectorsunitasknownkji
kVjViVV zyx
ˆ,ˆ,ˆ
ˆˆˆ
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Unit Vectors
• What are unit vectors?– Unit vectors have a magnitude of 1 and point along major axes
of our coordinate system
• Writing a vector with unit vectors is equivalent to multiplying each unit vector by a scalar
kVjViVV
kVVjVViVV
zyx
zzyyxx
ˆˆˆ
ˆ,ˆ,ˆ
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Unit Vectors
• For a vector with components:
• Write this in unit vector notation
2,3,4 zyx VVV
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Example: Vector Addition/Subtraction• Displacement
– A hiker traces her movement along a trail. The first leg of her hike brings her to the foot of the mountain:
– On the second leg, she ascends the mountain, which she figures to be a displacement of:
– On the third, she walks along a plateau.
– Then she falls of a cliff
– What is the hiker’s final displacement?
jmimV ˆ500ˆ25001
kmjmimV ˆ700ˆ700ˆ5002
jmV ˆ6003
kmV ˆ5004
Department of Physics and Applied Physics95.141, F2010, Lecture 3
Example: Vector Addition/Subtraction
jmimV ˆ500ˆ25001
kmjmimV ˆ700ˆ700ˆ5002
jmV ˆ6003
kmV ˆ5004
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