Chapter 3 Motion in Two and Three Dimensions; Vectors.

Chapter 3

Motion in Two and Three Dimensions; Vectors

Vectors and Scalars

A vector has magnitude as well as direction.

Some vector quantities: displacement, velocity, force, momentum

A scalar has only a magnitude.

Some scalar quantities: mass, time, temperature

Addition of Vectors – Graphical Methods

Adding the 2 vectors

Subtracting 2 vectors

Addition of Vectors – Graphical MethodsConsider a motion in two dimensions. Suppose, you move to the right 8.0 km and then 4.0 km up. What is your displacement?

By using the Pythagorean Theorem, we have

km

kmkmDR9.8

)4()8( 22

Adding Vectors by Components

Any vector can be expressed as the sum of two other vectors, which are called its components. Usually the other vectors are chosen so that they are perpendicular to each other.

Adding Vectors by Components

The components are found using trigonometric functions.

222

tan

sin

cos

yx

x

y

y

x

VVV

V

V

VV

VV

Adding Vectors by Components

Adding vectors:

1. Draw a diagram

2. Choose x and y axes.

3. Resolve each vector into x and y components.

4. Calculate each component using sines and cosines.

5. Add the components in each direction.

6. To find the length and direction of the vector, use:

Adding Vectors by Components

yyy

xxx

VVV

VVV

21

21

Components:

Resultant:

Direction:

Vector V1 is 6.6 units long and points along the negative x axis. Vector V2 is 8.5 units long and points at an angle of 45o to the positive x axis. (a) What are the x and y components of each vector? (b) Determine the sum (magnitude and angle).

(b)

The sum has a magnitude of 6.0 units, and is 84o clockwise from the – negative x-axis, or 96o counterclockwise from the positive x-axis.

2 2 1 o6.00.6 6.0 6.0 units tan 84

0.6 1 2V + V

V =

unitsVVV

unitsVVV

yyy

xxx

0.60.60

6.00.66.6

21

21

1 6.6 unitsxV 8.(a)1 0 unitsyV

o

2 8.5cos 45 6.0 unitsxV o

2 8.5sin 45 6.0 unitsyV

Unit vectors i and jExpress each vector as the sum of 2 perpendicular vectors. It is common to use the horizontal and vertical directions using unit vectors i and j

Example: A = Ax + Ay= Axi + Ayj , where Ax and Ay are the horizontal and vertical components

B = Bx + By = Bxi + Byj, where Bx and By are the horizontal and vertical components

Some useful properties of unit vectors

i.j=0; i.i=1; j.j=1

A.B = AxBx+AyBy

R = A+ B + C = i-2j

Three vectors are expressed as A = 4i – j, B = -3i + 2j, and C = -3j. If R = A+ B + C, find the magnitude and direction of R.

magnitude = 2.24

angle = 63.4 o, below the x-axis

Projectile Motion

A projectile is an object moving in two dimensions under the influence of Earth's gravity; its path is a parabola.

Projectile Motion

This photograph shows two balls that start to fall at the same time. The one on the right has an initial speed in the x-direction. It can be seen that vertical positions of the two balls are identical at identical times, while the horizontal position of the yellow ball increases linearly.

A projectile can be understood by analyzing the horizontal and vertical motions separately.

Projectile Motion

The speed in the x-direction is constant

in the y-direction the object moves with constant acceleration g.

Projectile Motion

If an object is launched at an initial angle of θ0 with the horizontal, the analysis is similar except that the initial velocity has a vertical component.

Solving Problems Involving Projectile Motion

Projectile motion is motion with constant acceleration in two dimensions, where the acceleration is g and is down.

A diver running 1.8 m/s dives out horizontally from the edge of a vertical cliff and 3.0 s later reaches the water below. How high was the cliff, and how far from its base did the diver hit the water?

Choose downward to be the positive y direction. The origin will be at the point where the diver dives from the cliff. In the horizontal direction,

0 1.8 m sxv and 0xa

In the vertical direction, 0 0yv 29.80 m sya 0 0y

and the time of flight is t = 3.0 s

The height of the cliff is found from applying to the vertical motion.

22 21 10 0 2 2

0 0 9.80 m s 3.0s 44 my yy y v t a t y

The distance from the base of the cliff to where the diver hits the water is found from the horizontal motion at constant velocity:

1.8 m s 3 s 5.4 mxx v t

A football is kicked at ground level with a speed of 18.0 m/s at an angle of 35.0º to the horizontal. How much long later does it hit the ground?

and the final y velocity will be the opposite of the starting y velocity. to find the time of flight use

o o

0

0 2

18.0sin 35.0 m s 18.0sin 35.0 m s 2.11 s

9.80 m sy y

y y

v vv v at t

a

Choose the point at which the football is kicked the origin, and choose upward to be the positive y direction. When the football reaches the ground again, the y displacement is 0. For the football,

o

0 18.0sin 35.0 m syv

29.80 m sya

35.0 o

V0

Vx0

Vy0

- Vy0

Vx0

Extra Slides

19. Apply the range formula

2

0 0

2

0 22

0

1 o o

0 0

sin 2

2.0 m 9.8 m ssin 2 0.4239

6.8 m s

2 sin 0.4239 13 ,77

vR

g

Rg

v

A fire hose held near the ground shoots water at a speed of 6.8 m/s. At what angle(s) should the nozzle point in order that the water land 2.0 m away ? Why are there two different angles? Sketch the two trajectories.

There are two angles because each angle gives the same range. If one angle is

o45 , then o45

is also a solution. The two paths are shown in the graph.

-0.5

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2

Addition of Vectors – Graphical Methods

Even if the vectors are not at right angles, they can be added graphically by using the “tail-to-tip” method.

Addition of Vectors – Graphical Methods

The parallelogram method may also be used;

Subtraction of Vectors, and Multiplication of a Vector by a Scalar

In order to subtract vectors, we define the negative of a vector, which has the same magnitude but points in the opposite direction.

Then we add the negative vector:

Subtraction of Vectors, and Multiplication of a Vector by a Scalar

A vector V can be multiplied by a scalar c; the result is a vector cV that has the same direction but a magnitude cV. If c is negative, the resultant vector points in the opposite direction.

Addition of Vectors – Graphical Methods

Adding the vectors in the opposite order gives the same result: