Kinematics in Two Dimensions;

Vectors

!! Scalars They are specified only by a number and units and have no direction associated with them, such as time, mass, and temperature.

!! Vectors

They have magnitude (a number & units) as well as direction. Examples of vector quantities are: displacement, velocity, and momentum.

Vectors & Scalars

When we write the symbol for a vector, we will use an arrow over the symbol, or a boldface type or with a tiny arrow over the symbol . If we are concerned only with the magnitude of a vector, we will simply write it in italics, as we do for other symbols.

Thus for velocity we write , and for it’s magnitude only we write v. !v

Addition of Vectors – Graphical Method

You do need to be careful about the signs, as the figure indicates.

Because vectors are quantities that have direction as well as magnitude, they must be added in a special way.

For vectors in one dimension, simple addition and subtraction are all that is needed.

8 km + 6 km = 14 km east of the point of origin

8 km - 6 km = 2 km east of the point of origin

If the motion is in two dimensions, i.e., the two vectors are not along the same line, the situation is somewhat more complicated and simple arithmetic cannot be used.

Addition of Vectors – Graphical Methods

For example, when a person walks 10.0 km east and then walks 5.0 km north, the actual travel paths are at right angles to one another.

We can find the displacement by using the Pythagorean Theorem when the vectors are perpendicular to each other.

!DR =

!D1 +

!D2

!DR = DR = D1

2 + D22 = 10.0 km( )2 + 5.0 km( )2

DR = 125 km2 = 11.2 km

Resultant Displacement :

= (11.2!km,!27°!N!of!E) !DR =

!D1 +

!D2

①! On a diagram, draw one of the vectors to scale.

②! Draw the second vector to scale, placing its tail at the tip of the first vector and being sure its direction is correct.

③! Draw the 3rd vector to scale, placing its tail at the tip of the second vector and being sure its direction is correct.

④! The arrow drawn from the tail of the first vector to the tip of the third vector represents the sum or the Resultant Vector.

General Rules for Adding Vectors

!VR =

!V1 +!V2 +

!V3

Adding Vectors: Parallelogram Method

Another way to add two vectors is the parallelogram method. It is equivalent to the tail-to-tip method. In this method, the two vectors are drawn starting from a common origin, and a parallelogram is constructed using these two vectors as adjacent sides. The resultant is the diagonal drawn from the common origin. The figure shows the tail-to-tip method as well, and we can see that both methods yield the same result.

Example (1)

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

Example (2)

Given that A + B = C, and that |A|2 +|B|2 =|C|2, 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 and in the opposite direction d)! They are at 45o to each other e)! They can be at any angle to each other

Subtraction of Vectors & 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:

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.

Resolving a Vector into Components

Any vector can be expressed as the sum of two other vectors, which are called its components of the original vector. Usually the other vectors are chosen to be along two perpendicular directions, such as the x and y axes. The process of finding the components is known as resolving the vector into its components.

!V =!Vx +

!Vy

!V Vx ,Vy( )

Adding Vectors by Components

To add vectors using the method of components, we need to use the trigonometric functions sine, cosine, and tangent.

Given any angle between two vectors, ", a right triangle can be constructed by drawing a line perpendicular to one of its sides.

The longest side of a right triangle, opposite the right angle, is called the hypotenuse, which we label h. The side opposite the angle " is labeled o, and the side adjacent is labeled a. We let h, o, and a represent the lengths of these sides, respectively.

Adding Vectors by Components We now define the three trigonometric functions, sine, cosine, and tangent (abbreviated sin, cos, tan), in terms of the right triangle, as follows:

sin! = Side oppositeHypotenuse

= oh

cos! = Side adjacentHypotenuse

= ah

tan! = Side oppositeSide adjacent

= oa

If the components are perpendicular, they can be found using trigonometric functions.

sin! =VyV

cos! = VxV

tan! =VyVx

Vy =V sin!Vx =V cos!sin2! + cos2! = 1

"

#$

%$&V 2 =Vx

2 +Vy2

sin2! + cos2! = o2

h2+ a

2

h2= o

2 + a2

h2= h

2

h2= 1

Specifying Vectors There are two ways to specify a vector in a given coordinate system:

"!You know the components, Vx and Vy

Vx =V cos!Vy =V sin!

!"#$V 2 =Vx

2 +Vy2

"!You know the magnitude of V and the angle " it makes with the positive x axis

V = Vx2 +Vy

2

tan! =VyVx

Example (3)

If each component of a vectors 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

Adding Vectors ①! Draw a diagram; add the vectors graphically. 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! Find the length and direction of the vector.

The components are effectively one-dimensional, so they can be added arithmetically

!VR =

!V1 +!V2!

VRx =!V1x +

!V2x!

VRy =!V1y +

!V2y

V = Vx2 +Vy

2 , tan! =VyVx

V1x =V1 cos!1 , V1y =V1 sin!1

VRx =V1x +V2x +! , VRy =V1y +V2y +!

Example (4)

An airplane trip involves three legs, with two stopovers. The first leg is due east for 620 km; the second leg is southeast (45°) for 440 km; and the third leg is at 53° south of west, for 550 km. What is the plane's total displacement?

Approach to Solve Example (4)

①! Draw a diagram: ②! Choose the axes ③! Resolve components ④! Calculate each component ⑤! Add the components ⑥! Find the magnitude and direction

Example (5)

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)! 30o

b)! 180o

c)! 90o

d)! 60o

e)! 45o

Projectile Motion A projectile is an object moving in two dimensions, near the Earth’s surface, and under the influence of Earth’s gravity. The path is a parabola.

Studying Projectile Motion Projectile motion can be understood by analyzing the horizontal and vertical components of the motion separately.

Example: the projectile of small ball falling from the edge of a table. Assume: -- the motion of the ball begins at time t = 0 -- the origin of an xy coordinate system is at x0 = y0 = 0 -- the initial velocity of the ball is vx0 . The dashed black line represents the path of the ball. The velocity vectors are solid green arrows, and velocity components are dashed green arrows.

(A vertically falling object starting from rest at the same place and time is shown at the left for comparison; vy is the same at each instant for the falling object and the projectile.)

Studying Projectile Motion At t = 0, ball has only an x component of velocity. Once the ball leaves the table (at t = 0), it experiences a vertically downward acceleration due to gravity, g. Thus vy is initially zero (vy0 = 0) but increases continually in the downward direction (until the ball hits the ground).

Vy =Vy0 + aytVy0 = 0ay = !g

"

#$

%$&Vy = !gt

x = x0 +Vx0t +12axt

2

x0 = 0ax = 0

!

"

##

$

##

% x =Vx0t

y = y0 +Vy0t +12ayt

2

y0 = 0Vy0 = 0ay = !g

"

#

$$$

%

$$$

& y = ! 12gt 2

Example (6)

A kicked football leaves the ground at an angle "0 = 37.0° with a velocity of 20.0 m/s. Calculate (a) the maximum height, (b) the time of travel before the football hits the ground, and (c) how far away it hits the ground. Assume the ball leaves the foot at ground level, and ignore air resistance and rotation of the ball.

Example (7)

A projectile is launched from the ground at an angle of 30°. At what point in its trajectory does this projectile have the least speed?

a)! just after it is launched b)! at the highest point in its flight c)! just before it hits the ground d)! halfway between the ground and the highest point e)! speed is always constant

Horizontal Range

The total distance the football traveled in part (c) of Example (6), is called the horizontal range, R. It applies to a projectile that lands at the same level it started that is, yfinal = y0

y = y0

t =2vy0

g

!"#

$#% R = vx0t = vx0

2vy0

g&'(

)*+=

2vx0vy0

g= 2v0

2 sin!0 cos!0

g

R = v02 sin2!0

g, Only if y final = y0,- ./

Example (8)

The spring-loaded gun can launch projectiles at different angles with the same launch speed. At what angle should the projectile be launched in order to travel the greatest distance before landing?

a)! 30o

b)! 15o

c)! 45o

d)! 60o

e)! 75o

R = v02 sin2!0g

Projectile Motion is Parabolic In order to demonstrate that projectile motion is parabolic, we need to find y as a function of x by eliminating t between the two equations for horizontal and vertical motion. We ignore air resistance, assume that g is constant, and for simplicity we set x0 = y0 = 0

x = vx0t! t = xvx0

y = vy0t "12gt2

#

$%

&%! y =

vy0vx0

'()

*+,x " g

2v2x0

'()

*+,x2

y = Ax ! Bx2

This is the standard equation for a parabola where A and B are constants.

Relative Velocity How observations made in different frames of reference are related to each other?

Each velocity is labeled by two subscripts: the first refers to the object, the second to the reference frame in which it has this velocity. For example, suppose a boat heads directly across a river.

vBS is the velocity of the Boat with respect to the Shore frame

vWS is the velocity of the Water with respect to the Shore frame (this is the river current).

vBW is the velocity of the Boat with respect to the Water frame (boat's velocity relative to the shore if the water were still.)

!vBS =

!vBW + !vWSIt is often useful to remember that for any two objects or reference frames, A and B, the velocity of A relative to B has the same magnitude, but opposite direction, as the velocity of B relative to A:

!vAB = !!vBA