Download Lecture 4

Physics I
95.141
LECTURE 4
9/15/10
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Exam Prep Problem
• Vince Carter’s vertical leap is 43”.
– A) (10pts) With what initial vertical velocity does
Carter leave the ground?
– B) (10pts) What is his hang time?
– C) (10pts) Assuming Carter leaps straight up at t=0s
and lands at just after t=T, draw the vectors for
James’ displacement, velocity, and acceleration at:
•
•
•
•
•
i) The instant he leaves the ground (t=0s)
ii) t=T/4
iii) t=T/2
iv) t=3T/4
v) t=T
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Exam Prep Problem
• Vince Carter’s vertical leap is 43”.
– A) (10pts) With what initial vertical velocity does
James leave the ground?
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Exam Prep Problem
• Vince Carter’s vertical leap is 43”.
– B) (10pts) What is his hang time?
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Exam Prep Problem
• Vince Carter’s vertical leap is 43”.
– C) (10pts) Assuming Carter leaps straight up at t=0s
and lands at just after t=T, draw the vectors for
James’ displacement, velocity, and acceleration at:
t 0
  
y v a
t T4
  
y v a
t T2
  
y v a
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
t  3T 4
  
y v a
t T
  
y v a
Outline
• Lecture 3 Review
• Vector Kinematics
• Relative Motion
• What do we know?
–
–
–
–
Units/Dimensions/Measurement/SigFigs
Kinematic equations
Freely falling objects
Vectors
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Lecture 3 Review
• Freely falling body problems
– Batman’s bat-hook
• Scalars and Vectors
• Graphical description of vectors and vector
addition.
• Vector components
• Mathematical description of vector addition
(addition of components)
• Unit Vectors
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Vector Kinematics
• We can now do kinematics in more than one dimension
– This is helpful, because we live in a 3D world!
• We previously described displacement as Δx, but this
was for 1D, where motion could only be positive or
negative.
• In more than 1 dimension, displacement is a vector

v

r
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Vector Kinematics
x  x2  x1
y
  
r  r2  r1
x
Now, instead of describing displacement in
terms of either vertical or horizontal position, we
can talk about a displacement vector!
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Vector Kinematics
• In unit vectors, we can write the displacement
vector as:
• We can now rewrite our expression for average
velocity:
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Vector Kinematics
• Average velocity only tells part of the story
• Just like for motion in 1D, we can let Δt get smaller and smaller….
• Gives instantaneous velocity vector:

lim r

v
t  0 t
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Velocity Vector
• The magnitude of the
average velocity vector
is NOT equal to the
average speed.
• But the magnitude of the
instantaneous velocity
vector is equal to the
instantaneous speed at
that time
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Instantaneous Velocity (math)
• To find the instantaneous velocity, we can take
the derivative of the position vector with respect
to time:

r  x (t )iˆ  y (t ) ˆj  z (t )kˆ
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Example
• Say we are given the position of an object to be:

r (t )  (4t 2  1)iˆ  2e  t ˆj  sin( 2t )kˆ
• Can we find the velocity as a function of time?
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Acceleration Vector
• Average acceleration:
• Instantaneous acceleration
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Example Problem
• Imagine we are given the position of an object as a
function of time
– Find displacement at t=1s and t=3s
– Find velocity and acceleration as a function of time
– Find velocity and acceleration at t=3s

 


r (t )  4(m s )t  2(m s 2 )t 2 iˆ   3(m)  3(m s 3 )t 3 ˆj
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Example Problem
• Imagine we are given the position of an object as a
function of time
– Find displacement at t=1s and t=3s
– Find velocity and acceleration as a function of time
– Find velocity and acceleration at t=3s

 


r (t )  4(m s )t  2(m s 2 )t 2 iˆ   3(m)  3(m s 3 )t 3 ˆj
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Example Problem
• Imagine we are given the position of an object as a
function of time
– Find displacement at t=1s and t=3s
– Find velocity and acceleration as a function of time
– Find velocity and acceleration at t=3s
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Example Problem
• Let’s say we are told that a Force causes an object to
accelerate in the -y direction at 5m/s2. The object has an
initial velocity in the +x direction of 10m/s, and in the +y
direction of 15 m/s, and starts at the point (0,0).
– A) Give the initial velocity vector of the object
– B) Plot x(t) vs. t
– C) Plot y(t) vs. t
– D) Plot the object’s trajectory in the xy plane
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Example Problem
• Let’s say we are told that a Force causes an object to
accelerate in the -y direction at 5m/s2. The object has an
initial velocity in the +x direction of 10m/s, and in the +y
direction of 15 m/s, and starts at the point (0,0)..
20
vy
– A) Give the initial velocity vector of the object
vx
20
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Example Problem
• Let’s say we are told that a Force causes an object to accelerate in
the -y direction at 5m/s2. The object has an initial velocity in the +x
direction of 10m/s, and in the +y direction of 15 m/s, and starts at the
point (0,0).
– Before we solve B-D, let’s determine equations of motion
(METHOD I)
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Example Problem
• Let’s say we are told that a Force causes an object to accelerate in
the -y direction at 5m/s2. The object has an initial velocity in the +x
direction of 10m/s, and in the +y direction of 15 m/s, and starts at the
point (0,0).
– Before we solve B-D, let’s determine equations of motion
(METHOD II)
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Example Problem
• Let’s say we are told that a Force causes an object to accelerate in
the -y direction at 5m/s2. The object has an initial velocity in the +x
direction of 10m/s, and in the +y direction of 15 m/s, and starts at the
point (0,0).
x (t )  vox t  10t
– B) Plot x(t) vs t
x(t)
100
t(s)
10
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
time
x(t)
0
0m
1
10m
2
20m
5
50m
10
100m
Example Problem
• Let’s say we are told that a Force causes an object to accelerate in
the -y direction at 5m/s2. The object has an initial velocity in the +x
direction of 10m/s, and in the +y direction of 15 m/s, starts at (0,0).
– C) Plot y(t) vs t
1 2
y(m)
y (t )  v oy t  at  15t  2.5t 2
2
t(s)
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
time
y(t)
0
0m
1
12.5m
2
20m
3
22.5m
4
20m
5
12.5
10
-100
x (t )  10t
Example Problem
y(t )  15t  2.5t 2
D) Plot object trajectory
• Choose coordinate system
y( x)  1.5 x  .025 x 2
y(m)
x(m)
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
time
x(t)
y(t)
0
0m
0m
1
10m
12.5m
2
20m
20m
3
30m
22.5m
4
40m
20m
5
50
12.5
10
100
-100
Relative Velocity
• So far we have looked at adding displacement
vectors
• May also find situations where we need to add
velocity vectors
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Relative Velocity
• Two velocities:
5m/s
25m/s
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Relative Velocity
• In this case, our hero would presumably prefer not to be
decapitated by the bridge
• So we are interested in his velocity relative to the bridge
• He is on a train moving at +25 m/s relative to the bridge
• His velocity relative to the train is -5m/s
• So his velocity relative to the bridge is:
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Relative Velocity
• What is Kirk’s velocity when he hits the ground?
– Assume he leaps when car is moving 20m/s
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Relative Velocity (Example 1)
• Imagine you are on a barge floating down the river with the current


vriver  3 m s i
• You walk diagonally across the barge with a velocity

v walk on barge  2 m s iˆ  2 m s ˆj
• What is your velocity with respect to the water?
• With respect to the river bank?
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Another River Problem
• A boat’s speed in still water is 1.85m/s. If you want to
directly cross a stream with a current 1.2m/s, what
upstream angle should you take?
95.141, F2010, Lecture 4
Department of Physics and Applied Physics
Today We Learned….
• Vector kinematics
–
–
–
–
–
–
Displacement vector
Average velocity vector
Inst. Velocity vector
Average acceleration vector
Inst. Acceleration vector
Vector equations of motion
• Relative Velocity
95.141, F2010, Lecture 4
Department of Physics and Applied Physics