Chapter 3Motion in a Plane

Motion in a Plane Vector Addition Velocity Acceleration Projectile motion

Graphical Addition and Subtraction of VectorsA 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.

NotationVector: The magnitude of a vector:Scalar: m (not bold face; no arrow)The direction of vector might be 35 south of east; 20 above the +x-axis; or.

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 RGraphical Addition of Vectors

Vector Simulation

ExamplesTrig TableVector ComponentsUnit Vectors

Types of Vectors

Relative Displacement VectorsVector AdditionVector Subtractionis a relative displacement vector of point P3 relative to P2

Vector Addition via Parallelogram

Graphical Method of Vector Addition

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

Vector Components

Vector Components

Graphical Method of Vector Addition

Unit Vectors in Rectangular Coordinates

Vector Components in Rectangular Coordinates

Vectors with Rectangular Unit Vectors

Dot Product - ScalarThe dot product multiplies the portion of A that is parallel to B with B

Dot Product - ScalarThe dot product multiplies the portion of A that is parallel to B with BIn 2 dimensionsIn any number of dimensions

Cross Product - VectorThe cross product multpilies the portion of A that is perpendicular to B with B

In 2 dimensionsIn any number of dimensionsCross Product - Vector

Velocity

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

The instantaneous velocity is represented by the slope of a line tangent to the curve on the graph of an objects position versus time.A displacement over an interval of time is a velocity

Acceleration

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

A nonzero acceleration changes an objects state of motionThese have interpretations similar to vav and v.

Motion in a Plane with Constant Acceleration - ProjectileWhat 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

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

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

Example continued:

t (sec)x (meters)y (meters)0000.21.411.220.42.832.040.64.242.480.85.662.521.07.072.171.28.481.431.49.890.29

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:

Chart2

00

1.4141.218

2.8282.044

4.2422.478

5.6562.52

7.072.17

8.4841.428

9.8980.294

t (sec)

x,y (m)

Sheet1

txy

000

0.21.4141.218

0.42.8282.044

0.64.2422.478

0.85.6562.52

17.072.17

1.28.4841.428

1.49.8980.294

Sheet1

00

00

00

00

00

00

00

00

t (sec)

x,y (m)

Sheet2

Sheet3

Example continued:This is a plot of the y position versus x position for the object (its trajectory). The objects path is a parabola.

Chart3

0

1.218

2.044

2.478

2.52

2.17

1.428

0.294

x (m)

y (m)

Sheet1

txy

000

0.21.4141.218

0.42.8282.044

0.64.2422.478

0.85.6562.52

17.072.17

1.28.4841.428

1.49.8980.294

Sheet1

0

0

0

0

0

0

0

0

x (m)

y (m)

Sheet2

Sheet3

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:At t = 3 sec:CONSTANT

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

Example: How far does the arrow in the previous example land from where it is released?The arrow lands when y = 0.Solving for t:The distance traveled is:

SummaryAdding and subtracting vectors (graphical method & component method)VelocityAccelerationProjectile motion (here ax = 0 and ay = g)

Projectiles ExamplesProblem solving strategySymmetry of the motionDropped from a planeThe home run

Could incorporate personal response system questions from the College Physics by G/R/R 2E ARIS site (www.mhhe.com/grr), Instructor Resources: CPS by eInstruction, Chapter 3, Questions 10 and 11.