Ch 03b motion in a plane

MFMcGraw-PHY 1401 Chapter 3b - Revised: 6/7/2010 1

Motion in a Plane

Chapter 3

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Motion in a Plane

• Vector Addition

• Velocity

• Acceleration

• Projectile motion

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Graphical Addition and Subtraction of Vectors

A vector is a quantity that has both a magnitude and a direction. Position is an example of a vector quantity.

A scalar is a quantity with no direction. The mass of an object is an example of a scalar quantity.

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Notation

Vector: FF

or

The magnitude of a vector: .or or FF

F

Scalar: m (not bold face; no arrow)

The direction of vector might be “35 south of east”; “20 above the +x-axis”; or….

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To add vectors graphically they must be placed “tip to tail”. The result (F1 + F2) points from the tail of the first vector to the tip of the second vector.

This is sometimes called the resultant vector R

F1

F2

R

Graphical Addition of Vectors

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Vector Simulation

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Examples

• Trig Table

• Vector Components

• Unit Vectors

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Types of Vectors

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Relative Displacement Vectors

C = A + B

C - A = B

Vector Addition

Vector Subtraction

B

is a relative displacement vector of point P3 relative to P2

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Vector Addition via Parallelogram

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Graphical Method of Vector Addition

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Think of vector subtraction A B as A+(B), where the vector B has the same magnitude as B but points in the opposite direction.

Graphical Subtraction of Vectors

Vectors may be moved any way you please (to place them tip to tail) provided that you do not change their length nor rotate them.

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Vector Components

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Vector Components

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Graphical Method of Vector Addition

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Unit Vectors in Rectangular Coordinates

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Vector Components in Rectangular Coordinates

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x y z A = A i + A j+ A kˆ ˆ ˆ

x y z B = B i +B j + B kˆ ˆ ˆ

Vectors with Rectangular Unit Vectors

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Dot Product - Scalar

The dot product multiplies the portion of A that is parallel to B with B

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Dot Product - Scalar

The dot product multiplies the portion of A that is parallel to B with B

In 2 dimensions

In any number of dimensions

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Cross Product - Vector

The cross product multpilies the portion of A that is perpendicular to B with B

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x y z

x y z

i j kA A AB B B

ˆ ˆ ˆ

y z z y

x z z x

x y x y

= (A B - A B )i

+ (A B - A B ) j

+ (A B - A B ) k

ˆˆˆ

A B = A B sin( )

In 2 dimensions

In any number of dimensions

Cross Product - Vector

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Velocity

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y

x

ri rf

t

rvav Points in the direction of r

r

vi

The instantaneous velocity points tangent to the path.vf

A particle moves along the curved path as shown. At time t1 its position is ri and at time t2 its position is rf.

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tt

rv lim0

velocityousInstantane

The instantaneous velocity is represented by the slope of a line tangent to the curve on the graph of an object’s position versus time.

t

rvav velocityAverage

txv x,av :be wouldcomponent - xThe

A displacement over an interval of time is a velocity

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Acceleration

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y

x

vi

ri rf

vf

A particle moves along the curved path as shown. At time t1 its position is r0 and at time t2 its position is rf.

v

Points in the direction of v.t

vaav

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t

vaavonaccelerati Average

A nonzero acceleration changes an object’s state of motion

Δt 0

ΔvInstantaneous acceleration = a = limΔt

These have interpretations similar to vav and v.

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Motion in a Plane with Constant Acceleration - Projectile

What is the motion of a struck baseball? Once it leaves the bat (if air resistance is negligible) only the force of gravity acts on the baseball.

Acceleration due to gravity has a constant value near the surface of the earth. We call it g = 9.8 m/s2

Only the vertical motion is affected by gravity

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The baseball has ax = 0 and ay = g, it moves with constant velocity along the x-axis and with a changing velocity along the y-axis.

Projectile Motion

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Example: An object is projected from the origin. The initial velocity components are vix = 7.07 m/s, and viy = 7.07 m/s.

Determine the x and y position of the object at 0.2 second intervals for 1.4 seconds. Also plot the results.

2f i iy y

f i ix

1Δy = y - y = v Δt + a Δt2

Δx = x - x = v Δt

Since the object starts from the origin, y and x will represent the location of the object at time t.

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t (sec) x (meters) y (meters)0 0 0

0.2 1.41 1.220.4 2.83 2.040.6 4.24 2.480.8 5.66 2.521.0 7.07 2.171.2 8.48 1.431.4 9.89 0.29

Example continued:

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0

2

4

6

8

10

12

0 0.5 1 1.5

t (sec)

x,y

(m)

This is a plot of the x position (black points) and y position (red points) of the object as a function of time.

Example continued:

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Example continued:

0

0.5

1

1.5

2

2.5

3

0 2 4 6 8 10

x (m)

y (m

)

This is a plot of the y position versus x position for the object (its trajectory). The object’s path is a parabola.

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Example (text problem 3.50): An arrow is shot into the air with = 60° and vi = 20.0 m/s.

(a) What are vx and vy of the arrow when t = 3 sec?

The components of the initial velocity are:

m/s 3.17sinm/s 0.10cos

iiy

iix

vvvv

At t = 3 sec:m/s 1.12

m/s 0.10

tgvtavv

vtavv

iyyiyfy

ixxixfx

x

y

60°

vi

CONSTANT

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(b) What are the x and y components of the displacement of the arrow during the 3.0 sec interval?

y

x

r

2x f i ix x ix

2 2y f i iy y iy

1Δr = Δx = x - x = v Δt + a Δt = v Δt + 0 = 30.0 m21 1Δr = Δy = y - y = v Δt + a Δt = v Δt - gΔt = 7.80 m2 2

Example continued:

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Example: How far does the arrow in the previous example land from where it is released?

The arrow lands when y = 0. 021 2 tgtvy iy

Solving for t: sec 53.32

gv

t iy

The distance traveled is: ixΔx = v Δt = 35.3 m

iy1Δy = (v - gΔt) t = 02

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Summary

• Adding and subtracting vectors (graphical method & component method)

• Velocity

• Acceleration

• Projectile motion (here ax = 0 and ay = g)

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Projectiles Examples

• Problem solving strategy

• Symmetry of the motion

• Dropped from a plane

• The home run