Ch 3 - Vectors and two dimensional motion

PPT 32 Special equations

- The motion equations can be combined algebraically and solved for the range and the maximum height (see pic)

PPT 2 A quantity that has both magnitude (size) and direction is?

A vector

PPT 3 Two vectors are equal if they have the same magnitude and the same direction.... this refers to which property of vectors?

Equality of two vectors propert

PPT 1 Why will it be more important to use vectors in this chapter?

For more complex motion, manipulating vectors will be more important

Book noted on projectile motion at various angles...

In general, the equations of constant acceleration developed in Chapter 2 follow separately for both the x-direction and the y-direction. An important difference is that the initial velocity now has two components, not just one as in that chapter. We assume that at t 5 0 the projectile leaves the origin with an initial velocity vS0. If the velocity vector makes an angle u0 with the horizontal, where u0 is called the projection angle, then from the definitions of the cosine and sine functions and Figure 3.14 we have

PPT 8 What do we do if we have many vectors?

Just keep repeating the "tip to tail" process until all are includes

PPT 38 Problem--solving strategy for relative velocity

LABEL: all the objects with a descriptive letter LOOK: For phrases such as "velocity of A relative to B" - write the velocity variable with appropriate notation - If there is something not explicitly noted as being relative to something else, it is probably relative to the earth TAKE: The velocities and put them into an equation - keep the subscripts in an order analogous to the standard equation SOLVE: for the unknowns

PPT 2 A scalar quantity has only what?

Magnitude (size)

PPT 2 A vector has both ___________ and ______________

Magnitude (size) and direction

PPT 9 More properties of vectors

NEGATIVE VECTORS: - The negative vector is defined as the vector that gives zero when added to the original vector - two vectors are negative if they have the same magnitude but are 180 degrees apart (opposite directions)

PPT 19 What is an important special case of two dimensional motion that we will be dealing with?

Projectile motion

PPT 11 The magnitude of the vector is multiplied or divided by the ?

Scalar

PPT 2 Does a Vector or a scalar quantity have only magnitude?

Scalar quantities have only magnitude

PPT 4 When using the geometric methods to add vectors, what can we use a tool?

Scale drawings

PPT 23 Some details about the rules

See the pic

PPT 18 Acceleration

The average acceleration is defined as the rate at which the velocity changes * see pic for eq. - The instantaneous acceleration is the limit of the average acceleration as delta t approaches zero -SI uni: M/s^2

PPT 7 A+B=B+A describes which law of vectors?

The commutative law of addition

PPT 37 Relative velocity equations

The rate of change of the displacements gives the relationship for the velocities

PPT 4 What is represented by an R with an arrow on top of it?

The resultant vector (sum)

PPT 4 When adding vectors, what must be the same?

The units must be the same

PPT 4 What must be taken into account when adding vectors?

Their directions must be taken into account

PPT 17 What is being described by the following statement "the limit of the average velocity as delta t approaches zero" Also, it is along a line that is tangent to the path of the particle and in the direction of motion

These describe instantaneous velocity

PPT 20 In the rules of projectile motion: The initial velocity can be broken down into its

X and y components

PPT 4 Can we add vectors algebraically?

Yes

PPT 2 When dealing with just the magnitude of a vector in print, a what is used?

an Italic letter

PPT 20 In the rules of projectile motion: The y direction is

free fall

PPT 20 In the rules of projectile motion: The x and y direction of motion are completely...

independent of each other

PPT 2 Vectors Vs Scalar Review All physical quantities encountered in this text will be either ....

scalar or vector

PPT 16 The change in the position of an object is?

the displacement *see pic for formula

PPT 20 In the rules of projectile motion: The x direction is ...

uniform motion

PPT 1 Were we able to use vectors in one dimensional motion?

yes, but only to a limited extent

PPT 36 RELATIVE position example

•The position of car A relative to car B is given by the vector subtraction equation

PPT 25 Velocity of the projectile

•The velocity of the projectile at any point of its motion is the vector sum of its x and y components at that point • • -Remember to be careful about the angle's quadrant •

PPT 35 Relative position equations

* see pic

PPT 3 Properties of Vectors

- EQUALITY OF TWO VECTORS: Two vectors are equal if the have the same magnitude and the same direction -MOVEMENT OF VECTORS IN A DIAGRAM - Any vector can be moved parallel to itself without being affected

PPT 1 Vectors and two dimensional motion

- In one-dimensional motion vectors were used to a limited extent - for more complex motion, manipulating vectors will be more important

PPT 33 Relative velocity

- Relative velocity is about relating the measurements of two different observers - It may be useful to use a moving frame of reference instead of a stationary one - It is important to specify the frame of reference, since the motion may be different in different frames of reference - There are no specific equations to learn to solve relative velocity problems

PPT 28 Problem solving strategy for projectile motion

- SELECT a coordinate system and skethe path of the projectile: include initial and final positions, velocities, and acceleration -RESOLVE: The initial velocity into y- and y-components -TREAT: the horizontal and vertical motions independently -FOLLOW: the techniques for solving problems with constant velocity to analyze the horizontal motion of the projectile -FOLLOW: The techniques for solving problems with constant acceleration to analyze the Vertical motion of the projectile

PPT 13 Components of a Vector, cont.

- The X-component of a vector is the projection along the x-axis *see pic for eq. - The y-component of a vector is the projection along the y-axis - Then A= Ax+Ay

PPT 17 Velocity

- The average velocity is the ratio of the displacement tot the time interval for the displacement * see pic for formula - The instantaneous velocity is the limit of the average velocity as the delta t approaches zero: The direction of the instantaneous velocity is along a line that is tangent to the path of the particle and in the direction of motion - Si unit: meter per second

PPT 16 Displacement

- The position of an object is described by its position vector r (with arrow on top) -The displacement of the object is defined as the change in its position * see pic - we use Si units: meter

PPT 27 Projectile motion summary, cont

- The vertical component of the Velocity Vy and the displacement in the y-direction are identical to those of a freely falling body - projectile motion can be described as a superposition of two independent motion in the x- and y-directions

PPT 21 Projectile motion.... a visual example

- The y component of velocity is zero at the peak of the path - The x component of velocity remains constant in time

PPT 30 Some Variations of projectile motion

- an object may be fired horizontally - the initial velocity is all in the x direction (see equation in the picture) - All the general rules of projectile motion apply

PPT 5 Adding vectors geometrically (triangle or polygon method)

- choose a scale -Draw the first vector with the appropriate length and in the direction specified with respect to a coordinate system -Draw the next vector using the same scale with the appropriate length and in the direction specified, with respect to a coordinate system whose origin is the end of the first vector and parallel to the coordinate system used for the first vector

PPT 22 Projectile motion at various initial angle

- complementary values of the initial angle result in the same range... the heights will be different - The maximum range occurs at a projection angle of 45 degree

PPT 5 Graphically adding vectors, cont'

- continue drawing the vectors "tip to tail" - the resultant is drawn from the origin of the first vector to the end of the last vector - measure the length of the resultant and its angle: use the scale factor to convert length to actual magnitude - This method is called the triangle method

PPT 31 Non-symmetrical projectile motion

- follow the general rules for the projectile motion - Break the y-direction into parts (up and down) (symmetrical back to initial height and then the rest of the height)

PPT 12 Components of a vector

- it is useful to use rectangular components to add vectors: These are the same projections of the vector a along the X- and Y axis

PPTT 26 Projectile motion summary

- provided air resistance is negligible, the horizontal component of the velocity remains constant since ax=0 -The vertical component of the acceleration is equal to the free fall acceleration -g -- the acceleration in the y-direction is not zero at the top of the projectile's trajectory

PPT 10 Vector Subtraction

- special case of vector addition: Add the negative of the subtracted vector - continue with standard vector addition procedure

PPT 14 More amount the components of a vector, cont.

- the components are the legs of the Right triangle whose hypotenuse is A (arrow on top) * see pic for the equations - may still have to find theta with respect to the positive x-axis -The value will be correct only if the angle lies in the first or fourth quadrant -In the second or third quadrant, add 180 degree

PPT 34 Relative Velocity notation

- the pattern of subscripts can be useful in solving relative velocity problems ASSUME the following notation: -- E is an observer, stationary with respect to the earth --A and B are two moving cars

PPT 11 Multiplying or dividing a vector by a scalar

- the result of the multiplication or division is a vector - the magnitude of the vector is multiplied of divided by the scalar - If the scalar is positive, the direction of the result is the same as the original vector - if the scalar is negative, the direction of the result is the opposite that of the original vector

PPT 20 Rules of projectile motion

- the x and y directions of motion are completely independent of each other - the x-direction is uniform motion * see pic for eq. - the y direction is free fall *see pic for formula - The initial velocity can be broken down into its x and y components *see pic for eq.

PPT 15 Motion in two dimensions.... 1. What can we use to describe motion in more than one dimension when + or - ar not sufficient? 2. Are we still interested in displacement, velocity, and acceleration?

- using a + or - sign is not always sufficient to fully describe motion in more than one dimension: Vectors can be used to more fully describe motion - Still interested in displacement, velocity, and acceleration

PPT 4 Adding Vectors

- when adding vectors, their directions must be taken into account -units must be the same - geometric methods: use scale drawings -Algebraic methods -The resultant vector sum is denoted as an R with an arrow on top of it R=A+B (all have arrows on top of them)

PPT 2 VECTOR NOTATION

- when handwritten, use an arrow: - When printed, will be in bold print with an arrow: - When dealing with just the magnitude of a vector in print, an italic letter will be used: A (italic) - Italics will also be used to represent scalars

PPT 8 Graphically adding vectors,cont

- when you have many vectors, just keep repeating the "tip to tail" process until all are included - The resultant is still drawn from the origin of the first vector to the end of the last vector

PPT 17 What is being described by the ratio of displacement to the time interval for the displacement

AVERAGE VELOCITY

PPT 19 Projectile motion

An object may move in both the X and y directions simultaneously: It move in two dimension - the form of the two dimensional motion we will deal with is an important special case called projectile motion

PPT 19 How else can we say that an object moves in two dimensions?

An object may move in both the x and y directions simultaneously

PPT 18 How is instantaneous acceleration defined?

As the limit of the average acceleration as delta t approaches zero

PPT 18 How is average acceleration defined?

As the rate at which the velocity changes

PPT 3 Any vector can be moved parallel to itself without being affected, is a property of vectors that describes?

The movement of vectors in a diagram

PPT 7 Vectors obey the commutative law of addition, what does this mean?

The order in which the vectors are added doesn't affect the result

PPT 11 The result of multiplication or division of a vector by a scalar is a ?

Vector

PPT 16 The displacement of the object is the

Vector

PPT 2 When handwritten, arrow must be used

Vector

PPT 2 When printed will be in bold print with an arrow

Vector

What is the difference in measurements of velocity and acceleration?

Velocity is measure m/s Acceleration measures in m/s^2

PPT 11 When multiplying or dividing a vector by a scalar, if the scalar is positive, the direction of the result if as the...

original vector

PPT 2 In addition to being used for vectors that deal with only magnitude, italics will also deal be used to reporesent

scalar quantities

PPT 11 When multiplying of dividing a vector by a scale, if the scalar is negative, the direction of the result is .....

the opposite of that of the original vector

PPT 9 How is a negative vector defined?

the vector that gives zero when added to the original vector

PPT 9 If two vectors are negative when

they have the same magnitude but are 180 degrees apart (opposite directions)