university physics: mechanics ch2. straight line motion lecture 3 dr.-ing. erwin sitompul

University Physics: Mechanics

Ch2. STRAIGHT LINE MOTION

Lecture 3

Dr.-Ing. Erwin Sitompulhttp://zitompul.wordpress.com

3/2Erwin Sitompul University Physics: Mechanics

Solution of Homework 2: Aprilia vs. KawasakiAn Aprilia and a Kawasaki are separated by 200 m when they start to move towards each other at t = 0. The Aprilia moves with initial velocity 5 m/s and acceleration 4 m/s2. The Kawasaki runs with initial velocity 10 m/s and acceleration 6 m/s2.

200 m0,apr 5m sv

2apr 4m sa

0,kwsk 10m sv 2

kwsk 6m sa

(a) Determine the point where the two motorcycles meet each other.

apr kwskx x1 12 2

0,apr 0,apr apr 0,kwsk 0,kwsk kwsk2 2x v t a t x v t a t

1 12 2

2 20 (5) (4) 200 ( 10) ( 6)t t t t

25 15 200 0t t


Solution of Homework 2: Aprilia vs. Kawasaki

2

1,2

15 (15) 4(5)( 200)

2(5)t

25 15 200 0t t

1 5 st • Possible answer• Both motorcycles meet

after 5 seconds• Where?

2 8 st

1 2apr 0,apr 0,apr apr2x x v t a t 1 2

kwsk 0,kwsk 0,kwsk kwsk2x x v t a t

1 2

20 (5)(5) (4)(5)

75 m

1 2

2200 ( 10)(5) ( 6)(5)

75 m

Thus, both motorcycles meet at a point 75 m from the original position of Aprilia or 125 m from the original position of Kawasaki.


Solution of Homework 2: Aprilia vs. Kawasaki

(b) Determine the velocity of Aprilia and Kawasaki by the time they meet each other.

apr 0,apr aprv v a t

(5) (4)(5)

25m s

kwsk 0,kwsk kwskv v a t

( 10) ( 6)(5)

40m s• What is the meaning of

negative velocity?


Free Fall Acceleration

Accelerating feathers and apples in a vacuum

A free falling object accelerates downward constantly (air friction is neglected).

The acceleration is a = –g = –9.8 m/s2, which is due to the gravitational force near the earth surface.

This value is the same for all masses, densities and shapes.

0

1 20 0 2

2 20 0

10 02

1 20 2

2 ( )

( )

v v at

x x v t at

v v a x x

x x v v t

x x vt at

With a = –g = –9.8 m/s2.

0

1 20 0 2

2 20 0

10 02

1 20 2

2 ( )

( )

v v gt

y y v t gt

v v g y y

y y v v t

y y vt gt


Example: Niagara Free Fall

In 1993, Dave Munday went over the Niagara Falls in a steel ball equipped with an air hole and then fell 48 m to the water. Assume his initial velocity was zero, and neglect the effect of the air on the ball during the fall.

(a) How long did Munday fall to reach the water surface?

0

1 20 0 2

2 20 0

10 02

1 20 2

2 ( )

( )

v v gt

y y v t gt

v v g y y

y y v v t

y y vt gt

1 20 0 2

y y v t gt

1 2

248 0 0 9.8t t

2 (48)(2)

9.8t 9.80

9.80 3.13 st

Niagara Falls




(b) What was Munday’s velocity as he reached the water surface?

0v v gt

2 20 02 ( )v v g y y

2 20 2(9.8)( 48 0)v 940.8 30.67 m/sv

0 (9.8)(3.13)v 30.67 m/s

or

940.8

0

1 20 0 2

2 20 0

10 02

1 20 2

2 ( )

( )

v v gt

y y v t gt

v v g y y

y y v v t

y y vt gt

• Positive (+) or negative (–) ?




(c) Determine Munday’s position and velocity at each full second.

1 20 0 2

0

y y v t gtv v gt

1 2

21 0 0 1 (9.8)(1) 4.9

0 (9.8)(1) 9.8t y

v

1 2

22 0 0 2 (9.8)(2) 19.6

0 (9.8)(2) 19.6t y

v

3t


Example: Egg Down the Bridge

Hanging over the railing of a bridge, you drop an egg (no initial velocity) as you also throw a second egg downward.

Which curves in the next figure give the velocity v(t) for (a) the dropped egg and (b) the thrown egg? (Curves A and B are parallel; so are C, D, and E; so are F and G.)

(a) D

(b) E


Short Lab: Determining Constant of Gravity

0

1 20 0 2

2 20 0

10 02

1 20 2

2 ( )

( )

v v gt

y y v t gt

v v g y y

y y v v t

y y vt gt

Preparation:1. Make a group of 3 students.2. Give name to your group.

Materials:1. An object that you can drop without destroying it2. A stopwatch (any handphone has it)

Procedure:1. Drop the object from an elevation the same

as the height of one of the group member.2. Measure the time from the release until the

object touches the floor.3. Repeat 5 times, average the result.4. Use the average fall time to calculate the

constant of gravity.5. Submit your group report.


Example: Baseball Pitcher

A pitcher tosses a baseball up along a y axis, with an initial speed of 12 m/s.

(a) How long does the ball take to reach its maximum height?

0v v gt 0 12 (9.8)t

1.22 s12

9.8t

0

1 20 0 2

2 20 0

10 02

1 20 2

2 ( )

( )

v v gt

y y v t gt

v v g y y

y y v v t

y y vt gt



(b) What is the ball’s maximum height above its release point?

0

1 20 0 2

2 20 0

10 02

1 20 2

2 ( )

( )

v v gt

y y v t gt

v v g y y

y y v v t

y y vt gt

1 20 0 2

y y v t gt 1 2

20 (12)(1.22) (9.8)(1.22)y

7.35 m




(c) How long does the ball take to reach a point 5.0 m above its release point?

1 20 0 2

y y v t gt 1 2

25 0 (12) (9.8)t t

24.9 12 5 0t t

1 20.53, 1.92t t

• Both t1 and t2 are correct• t1 when the ball goes up, t2 when the ball goes down


University Physics: Mechanics

Ch3. VECTOR QUANTITIES

Lecture 3

Dr.-Ing. Erwin Sitompulhttp://zitompul.wordpress.com


Vectors and Scalars

A displacement vector represents the change of position.

Both magnitude and direction are needed to fully specify a displacement vector.

The displacement vector, however, tells nothing about the actual path.

A vector quantity has magnitude as well as direction. Some physical quantities that are vector quantities are

displacement, velocity, and acceleration. A scalar quantity has only magnitude. Some physical quantities that are scalar quantities are

temperature, pressure, energy, mass, and time.


Adding Vectors Geometrically Suppose that a particle moves from

A to B and then later from B to C. The overall displacement (no matter

what its actual path) can be represented with two successive displacement vectors, AB and BC.

The net displacement of these two displacements is a single displacement from A to C.

AC is called the vector sum (or resultant) of the vectors AB and BC.

The vectors are now redrawn and relabeled, with an italic symbol and an arrow over it.

For handwriting, a plain symbol and an arrow is enough, such as a, b, s. s a b

→ → →


Adding Vectors Geometrically Two vectors can be added in either order.

a b b a

Three vectors can be grouped in any way as they are added.

( ) ( )a b c a b c


Adding Vectors Geometrically The vector –b is a vector with the same magnitude as b but

the opposite direction. Adding the two vectors would yield

( ) 0b b

Subtracting b can be done by adding –b.

( )d a b a b

d b a a d b or

→ →

→ →


Adding vectors geometrically can be tedious. An easier technique involves algebra and requires that the

vectors be placed on a rectangular coordinate system. For two-dimensional vectors, the x and y axes are usually

drawn in the drawing plane.

A component of a vector is the projection of the vector on an axis.

cosxa a sinya a

2 2x ya a a

tan y

x

a

a

| |a a

a

Components of Vectors


Note, that the angle θ is the angle that the vector makes with the positive direction of the x axis.

The calculation of θ is done in counterclockwise directions.

What is the value of θ on the figure above?

θ = 324.46° or –35.54°



Vectors with various sign of components are shown below:



Which of the indicated methods for combining the x and y components of vector a are proper to determine that vector?


→


Example: Flight in OvercastA small airplane leaves an airport on an overcast day and it is later sighted 215 km away, in a direction making an angle of 22° east of due north. How far east and north is the airplane from the airport when sighted?

Solution:90 22 68

cos (215 km)(cos68 )xd d 80.54 km

sin (215 km)(sin68 )yd d 199.34 km

Thus, the airplane is 80.54 km east and 199.34 km north of the airport.


Trigonometric Functions

Sine

Cosine

Tangent


Unit Vectors A unit vector is a vector that has a

magnitude of exactly 1 and points in a particular direction.

The unit vectors in the positive directions of the x, y, and z axes are labeled i, j, and k.

Unit vectors are very useful for expressing other vectors:

ˆ ˆi jx ya a a ˆ ˆi jx yb b b

^ ^ ^


Vector a with magnitude 17.0 m is directed +56.00 counterclockwise from the +x axis. What are the components ax and ay?

Example: Inclined Coordinate→

A second coordinate system is inclined 18.00 with respect to the first. What are the components of ax’ and ay’?

2 2 '2 '2x y x ya a a a a


Adding Vectors Geometrically Vectors can be added geometrically by subsequently linking

the head of one vector with the tail of the other vector. The addition result is the vector that goes from the tail of the

first vector to the head of the last vector.


Adding Vectors by Components Another way to add vectors is to combine their

components axis by axis.

x x xr a b y y yr a b

xaxb

ya

yb

xr

yr

xa xbya

yb

r a b ˆ ˆi jx yr r r

r

a

b r

r


Adding Vectors by Components

In the figure below, find out the components of d1, d2, d1+d2.

xr

yr

Component Notation

Magnitude-angle Notation

2 2x yr r r

tan y

x

r

r

where

r

ˆ ˆi jx yr r r

r r

→ → → →


Desert Ant

Adding Vectors by Components


TriviaA bear walks 5 km to the south. Resting a while, it continues walking 5 km east. Then, it walks again 5 km to the north and the bear reaches its initial position.What is the color of the bear?

5 km,south

5 km,north

5 kmeast

Polar Bear with White Fur


Homework 3: The BeetlesTwo beetles run across flat sand, starting at the same point. The red beetle runs 0.5 m due east, then 0.8 m at 30° north of due east. The green beetle also makes two runs; the first is 1.6 m at 40° east of due north. What must be (a) the magnitude and (b) the direction of its second run if it is to end up at the new location of red beetle?

1st run

2nd run

1st run

2nd run?


Homework 3A: Dora’s HouseDora, accompanied by Boots, wants to go back to her house, 10 km north east of their current position. To avoid the forest, Dora decides that they should first walk in the direction 20° north of due east, before later keep walking due north until reaching her house.(a) Determine the distance Dora and

Boots must travel in their first walk.

(b) Determine the distance they must travel in their second walk.

10 k

m

(c) Determine the total distance they must traveled to reach Dora’s house.


Homework 3B: Bank Robbers1. A blind man wants to go from Place A to Place B, which are separated by

250 m. Starting from Place A, the blind man walks 240 m in a direction 15° to the south of east direction and then walks 8 m to the north. (a) How far is the blind man from Place B after his second walk?(b) In what direction must he continue his walk so that he can reach

Place B?

2. A bank in downtown Boston is robbed. To elude police, the robbers escape by helicopter, making three successive flights described by the following displacements: 32 km, 45° south of east; 53 km, 26° north of west; 26 km, 18° east of south. At the end of the third flight they are captured. In what town are they apprehended?Hint: Reprint the map and draw the escape path of the robbers as accurate as possible using pencil.

university physics: mechanics ch2. straight line motion lecture 3 dr.-ing. erwin sitompul

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