Chapter 6

Motion InTwo-Dimensional

Motion in Two Dimensions Using ________signs is not always

sufficient to fully describe motion in more than one dimension. ______ can be used to more fully describe

motion. Still interested in displacement,

velocity, and acceleration. We can describe each of these in terms of

vectors.

Displacement The _______ of an

object is described by its position vector, .

The displacement of the object is defined as the change in its position.

r

fi r r r

Velocity The average velocity is the ratio of the

______________ to the ___________ for the displacement.

The instantaneous velocity is the limit of the average velocity as Δt 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.

av t

r

v

Acceleration The average acceleration is defined as

the rate at which the ______ changes.

The instantaneous acceleration is the limit of the average acceleration as Δt approaches zero.

av t

v

a

Unit Summary (SI)

Displacement m (All displacement has the dimensions of length,

L) Average velocity and instantaneous

velocity m/s (All velocities have the dimensions of L/T)

Average acceleration and instantaneous acceleration m/s2 (All accelerations have the dimensions of

L/T2.

Ways an Object Might Accelerate

The ____________of the velocity (the speed) can change

The ______________ of the velocity can change. Even though the magnitude is constant

_____the ________and the ________ can change.

Projectile Motion

An object may __________x and y directions ___________. It moves in two dimensions

The form of two dimensional motion we will deal with is an important special case called _____________.

Assumptions of Projectile Motion

We may ignore ___________. We may ignore the ___________________ Disregard the _________ of the earth. With these assumptions, an object in

projectile motion will follow a parabolic path.

Characteristics of Projectile Motion

The x- and y-directions of motion are completely _____________ of each other.

The x-direction is _____________: ax = 0

The y-direction is ____________: ay = -g

The _____________can be broken down into its ____________________: cos sinOx O O Oy O Ov v v v

Projectile Motion

Some Details About the Rules

x-direction: the horizontal motion ax = 0

This is constant throughout the entire motion. ∆x = voxt

This is the only operative equation in the x-direction since there is uniform velocity in that direction.

The _______________is dependent on the _______________and the _____________.

constantxx o o ov v v cosθ

More Details About the Rules

y-direction: the vertical motion Vy = 0 m/s (at height) Free fall problem

a = -g Take the positive direction as upward. Uniformly accelerated motion, so use the

____________________to solve for this motion.

yo o ov v sinθ

Velocity of the Projectile

The _________ of the projectile at any point of its motion is the _________of its x- and y-components at that point:

Remember to be careful about the angle’s quadrant.

2 2 1tan yx y

x

vv v v and

v

Projectile Motion Summary

Provided air resistance is negligible, the horizontal component of the velocity remains constant. Since ax = 0

The vertical component of the velocity vy is equal to the free fall acceleration –g The acceleration at the top of the

trajectory is not zero.

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 motions in the x- and y-directions.

The projection angle (θ) that the velocity vector makes with the x-axis is given by:

Θ = tan-1 vy/vx

Problem-Solving Strategy

Select a coordinate system and sketch the path of the projectile Include initial and final positions, velocities,

and accelerations Resolve the initial velocity into x- and

y-components Treat the horizontal and vertical

motions independently

Problem-Solving Strategy (cont.)

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.

Sample Problem

A long jumper leaves the ground at an angle of 20.0o to the horizontal and at a speed of 11.0 m/s. (a) How far does he jump? (Assume that his motion is equivalent to that of a particle.) (b) What maximum height is reached?

Circular Motion

Specific type of 2-dimensional motion.

Uniform circular motion: The movement of an

object or particle trajectory at a constant speed around a circle with a fixed radius.

Displacement During Circular Motion

As an object moves around a circle its ___________does not change lengths but its _________ does. To get the object’s

velocity we need the change in displacement.

Centripetal Acceleration

An object traveling in a circle, even though it moves with a constant speed, will have an acceleration.

The ____________ ____________ is due to the change in the ___________ of the velocity.

Centripetal Acceleration Centripetal refers

to “center-seeking.”

The direction of the velocity changes.

The acceleration is directed toward the ______ of the circle of motion.

Centripetal Acceleration The magnitude of the centripetal

acceleration is given by:

This direction is toward the center of the circle. So, whenever an object is moving in a circular

motion it has centripetal acceleration. Force causing this acceleration will be

discussed later.

2

c

va

r

Forces Causing Centripetal Acceleration

There are two things happening as an object moves along a circular path: Object wants to

move along a straight line.

Object wants to move along the circular path.

Which one wins out?

Forces Causing Centripetal Acceleration

Newton’s Second Law says that the centripetal acceleration is accompanied by a force. FC = maC

FC stands for any force that keeps an object following a circular path.

Tension in a string Gravity Force of friction

Centripetal Force

General equation:

If the force ___________, the object will move in a ____________tangent to the circle of motion.

Centripetal force is a classification that includes forces acting toward a central point. It is not a force in itself.

2

C C

mvF ma

r

Centripetal Force

In each example identify the centripetal force.

Sample Problem

A 13 g rubber stopper is attached to a 0.93 m string. The stopper is swung in a horizontal circle, making one revolution in 1.18 s. Find the tension force exerted by the string on the stopper.

Chapter 6

Motion InTwo-Dimensional

THE END