Lecture 4

Physics 207: Lecture 4, Pg 1

Lecture 4 Goals for Chapter 3 & 4

Perform vector algebra• (addition & subtraction) graphically or by xyz components • Interconvert between Cartesian and Polar coordinates

Begin working with 2D motion• Distinguish position-time graphs from particle trajectory plots• Trajectories

Obtain velocities Acceleration: Deduce components parallel and perpendicular to the trajectory path

Solve classic problems with acceleration(s) in 2D (including linear, projectile and circular motion)

Discern different reference frames and understand how they relate to motion in stationary and moving frames motion in stationary and moving frames

Assignment: Read thru Chapter 4 Assignment: Read thru Chapter 4

MP Problem Set 2 MP Problem Set 2 due this coming Wednesday


Welcome to Wisconsin

You are traveling on a two lane highway in a car going a speed of 20 m/s. You are notice that a deer that has jumped in front of a car traveling at 40 m/s and that car avoids hitting the deer but does so by moving into your lane! There is a head on collision and your car travels a full 2m before coming to rest. Assuming that your acceleration in the crash is constant. What is your acceleration in terms of the number of g’s (assuming g is 10 m/s2)?


Welcome to Wisconsin

You are traveling on a two lane highway in a car going a speed of 20 m/s (~44 mph). You are notice that a deer that has jumped in front of a car traveling at 40 m/s and that car avoids hitting the deer but does so by moving into your lane! There is a head on collision and your car travels a full 2m before coming to rest. Assuming that your acceleration in the crash is constant. What is your acceleration in terms of the number of g’s (assuming g is 10 m/s2)?

Draw a Picture Key facts (what is important, what is not important) Attack the problem


Welcome to Wisconsin You are traveling on a two lane highway in a car going a speed

of 20 m/s. You are notice that a deer that has jumped in front of a car traveling at 40 m/s and that car avoids hitting the deer but does so by moving into your lane! There is a head on collision and your car travels a full 2 m before coming to rest. Assuming that your acceleration in the crash is constant. What is the magnitude of your acceleration in terms of the number of g’s (assuming g is 10 m/s2)?

Key facts: vinitial = 20 m/s, after 2 m your v = 0.

x = xinitial + vinitial t + ½ a t2

x - xinitial = -2 m = vinitial t + ½ a t2

v = vinitial + a t -vinitial /a = t

-2 m = vinitial -vinitial /a + ½ a (-vinitial /a )2

-2 m= -½vinitial 2 / a a = (20 m/s) 2/2 m= 100m/s2


Vectors and 2D vector addition

The sum of two vectors is another vector.

A = B + CB

C A

B

C

D = B + 2 C ???


2D Vector subtraction

Vector subtraction can be defined in terms of addition.

B - C

B

C

B

-C

B - C

= B + (-1)C

A Different direction and magnitude !A = B + C


Reference vectors: Unit Vectors

A Unit Vector Unit Vector is a vector having length 1 and no units

It is used to specify a direction. Unit vector uu points in the direction of UU

Often denoted with a “hat”: uu = û

U = |U| U = |U| ûû

û û

x

y

z

ii

jj

kk

Useful examples are the cartesian unit vectors [ i, j, ki, j, k ] Point in the direction of the

x, y and z axes.

R = rx i + ry j + rz k


Vector addition using components:

Consider, in 2D, CC = AA + BB.

(a) CC = (Ax ii + Ay jj ) + (Bx i i + By jj ) = (Ax + Bx )ii + (Ay + By )

(b) CC = (Cx ii + Cy jj )

Comparing components of (a) and (b):

Cx = Ax + Bx

Cy = Ay + By

|C| =[ (Cx)2+ (Cy)2 ]1/2

CC

BxAA

ByBB

Ax

Ay


ExampleVector Addition

A. {3,-4,2}

B. {4,-2,5}

C. {5,-2,4}

D. None of the above

Vector A = {0,2,1} Vector B = {3,0,2} Vector C = {1,-4,2}

What is the resultant vector, D, from adding A+B+C?


ExampleVector Addition

A.A. {{3,-4,2}

B. {4,-2,5}

C. {5,-2,4}

D. None of the above

Vector A = {0,2,1} Vector B = {3,0,2} Vector C = {1,-4,2}

What is the resultant vector, D, from adding A+B+C?


Converting Coordinate Systems In polar coordinates the vector R = (r,)

In Cartesian the vector R = (rx,ry) = (x,y)

We can convert between the two as follows:

• In 3D cylindrical coordinates (r,z), r is the same as the magnitude of the vector in the x-y plane [sqrt(x2 +y2)]

tan-1 ( y / x )

22 yxr

y

x

(x,y)

rry

rx

j i

sin

cos

yxr

ryr

rxr

y

x


Resolving vectors into componentsA mass on a frictionless inclined plane

A block of mass m slides down a frictionless ramp that makes angle with respect to horizontal. What is its acceleration a ?

ma


Resolving vectors, little g & the inclined plane

g (bold face, vector) can be resolved into its x,y or x’,y’ components g = - g j g = - g cos j’ + g sin i’

The bigger the tilt the faster the acceleration…..

along the incline

x’

y’

x

yg


Dynamics II: Motion along a line but with a twist(2D dimensional motion, magnitude and directions)

Particle motions involve a path or trajectory

Recall instantaneous velocity and acceleration

These are vector expressions reflecting x, y & z motion

r = r(t) v = dr / dt a = d2r / dt2


Instantaneous Velocity But how we think about requires knowledge of the path. The direction of the instantaneous velocity is along a line that is

tangent to the path of the particle’s direction of motion.

v

The magnitude of the instantaneous velocity vector is the speed, s. (Knight uses v)

s = (vx2 + vy

2 + vz )1/2


Average Acceleration

The average acceleration of particle motion reflects changes in the instantaneous velocity vector (divided by the time interval during which that change occurs).

a The average

acceleration is a vector quantity directed along ∆v

( a vector! )


Instantaneous Acceleration

The instantaneous acceleration is the limit of the average acceleration as ∆v/∆t approaches zero

The instantaneous acceleration is a vector with components parallel (tangential) and/or perpendicular (radial) to the tangent of the path

Changes in a particle’s path may produce an acceleration The magnitude of the velocity vector may change The direction of the velocity vector may change

(Even if the magnitude remains constant) Both may change simultaneously (depends: path vs time)


Lecture 4

Assignment: Read through Chapter 4Assignment: Read through Chapter 4

MP Problem Set 2 MP Problem Set 2 due Wednesdaydue Wednesday

Lecture 4

Documents

Transcript of Lecture 4