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Chapter 3 Motion in Two Dimensions

© 2010 Pearson Education, Inc. 1

Units of Chapter 3 •  Components of Motion

•  Vector Addition and Subtraction

•  Projectile Motion

•  Relative Velocity


3.1 Components of Motion An object in motion on a plane can be located using two numbers—the x and y coordinates of its position. Similarly, its velocity can be described using components along the x- and y-axes.


3.1 Components of Motion

The velocity components are:

The magnitude of the velocity vector is:



The components of the displacement are then given by:

Note that the x- and y-components are calculated separately.



The equations of motion are:

When solving two-dimensional kinematics problems, each component is treated separately. The time is common to both.


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If two vectors are given such that A + B = 0, what can you say about the magnitude and direction of vectors A and B?

a) same magnitude, but can be in any direction

b) same magnitude, but must be in the same direction

c) different magnitudes, but must be in the same direction

d) same magnitude, but must be in opposite directions

e) different magnitudes, but must be in opposite directions

Question 3.1 Vectors

Given that A + B = C, and that lAl + lBl = lCl , how are vectors A and B oriented with respect to each other?

a) they are perpendicular to each other

b) they are parallel and in the same direction

c) they are parallel but in the opposite direction

d) they are at 45° to each other

e) they can be at any angle to each other

Question 3.1 Vectors

If each component of a

vector is doubled, what

happens to the angle of

that vector?

a) it doubles

b) it increases, but by less than double

c) it does not change

d) it is reduced by half

e) it decreases, but not as much as half

Question 3.1 Vector Components

A certain vector has x and y components

that are equal in magnitude. Which of the

following is a possible angle for this vector

in a standard x-y coordinate system?

a) 30°

b) 180°

c) 90°

d) 60°

e) 45°

Question 3.1 Vector Components


If the acceleration is not parallel to the velocity, the object will move in a curve:


3.2 Vector Addition and Subtraction

Geometric methods of vector addition

Triangle method:


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3.2 Vector Addition and Subtraction The negative of a vector has the same magnitude but is opposite in direction to the original vector. Adding a negative vector is the same as subtracting a vector.


3.2 Vector Addition and Subtraction Vector Components and the Analytical

Component Method

If you know A and B, here is how to find C:



The components of C are given by:

Equivalently,



Vectors can also be written using unit vectors:


3.2 Vector Addition and Subtraction Vectors can be resolved into components and the components added separately; then recombine to find the resultant.



This is done most easily if all vectors start at the origin.


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Question 3.2 Vector Addition

You are adding vectors of length

20 and 40 units. What is the only

possible resultant magnitude that

you can obtain out of the

following choices?

a) 0

b) 18

c) 37

d) 64

e) 100

3.3 Projectile Motion

An object projected horizontally has an initial velocity in the horizontal direction, and acceleration (due to gravity) in the vertical direction. The time it takes to reach the ground is the same as if it were simply dropped.


3.3 Projectile Motion A projectile launched in an arbitrary direction may have initial velocity components in both the horizontal and vertical directions, but its acceleration is still downward.


3.3 Projectile Motion The vertical motion is the same as if the object were thrown straight up or down with the same initial y velocity, and the horizontal velocity is constant.


3.3 Projectile Motion The range of a projectile is maximum (if there is no air resistance) for a launch angle of 45°.


3.3 Projectile Motion With air resistance, the range is shortened, and the maximum range occurs at an angle less than 45°.


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Question 3.3 Firing Balls I A small cart is rolling at constant velocity on a flat track. It fires a ball straight up into the air as it moves. After it is fired, what happens to the ball?

a) it depends on how fast the cart is moving

b) it falls behind the cart c) it falls in front of the cart d) it falls right back into the cart e) it remains at rest

Now the cart is being pulled along a horizontal track by an external force (a weight hanging over the table edge) and accelerating. It fires a ball straight out of the cannon as it moves. After it is fired, what happens to the ball?

a) it depends upon how much the track is tilted

b) it falls behind the cart

c) it falls in front of the cart d) it falls right back into the cart

e) it remains at rest

Question 3.3 Firing Balls II

You drop a package from a plane flying at constant speed in a straight line. Without air resistance, the package will:

a) quickly lag behind the plane while falling

b) remain vertically under the plane while falling

c) move ahead of the plane while falling

d) not fall at all

Follow-up: what would happen if air resistance is present?

Question 3.3 Dropping a Package Question 3.3 Shoot the Monkey I You are trying to hit a friend with a water balloon. He is sitting in the window of his dorm room directly across the street. You aim straight at him and shoot. Just when you shoot, he falls out of the window! Does the water balloon hit him?

Assume that the shot does have enough speed to reach the dorm

across the street.

a) yes, it hits

b) maybe—it depends on the speed of the shot

c) no, it misses d) the shot is impossible e) not really sure

3.4 Relative Velocity Velocity may be measured in any inertial reference frame. At top, the velocities are measured relative to the ground; at bottom they are measured relative to the white car.


3.4 Relative Velocity In two dimensions, the components of the velocity, and therefore the angle it makes with a coordinate axis, will change depending on the point of view.


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Review of Chapter 3 •  Two-dimensional motion is analyzed by considering each component separately. Time is the common factor.


Review of Chapter 3 •  Vector components:

•  In projectile motion, the horizontal and vertical motions are determined separately.


Review of Chapter 3

•  Range is the maximum horizontal distance traveled.

•  Relative velocity is expressed relative to a particular reference frame.


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