1Today: 1) Kinematics in two dimensions

Motion, Forces, Energy, Heat, Waves

Dr. David M. LindDr. Kun YangDr. David Van Winkle

Today:v Kinematics in two

dimensions kinematical equations vectors projectile motion preview: Newtons Laws

PHY 2053C College Physics AFall 2004

L4Ch3,4

Free Fall

Galileo Galilei, 1564-1642

Free Fall is one-dimensional motion with a constant, negativeacceleration. This means the acceleration is always pointing downward!

Galileo observed that all objects fall with the same, constantaccelerationg = -9.80 m/s2 (downward)

(as long as air-resistance is small)

2Introduction Projectile Motion

2 special cases of 1d-motion:v motion with constant

velocity (e.g. horizontal motion on a frictonless rail)

v motion with constant acceleration (e.g. free fall)

Projectile Motion is combination of horizontal constant velocityand vertical free fall

2d-MotionMotion in the horizontal and vertical directions are independent.

X and Y are independent (linked only by time).There is no acceleration in the X direction. There is acceleration in the Y direction.

Vertical Motion:constant acceleration,A = -9.80 m/s2

x t x0 vx0 tvx t vx0(const)

Horizontal Motion:constant velocity, zero acceleration

y t y0 vy0 t 1 2gt2

vy t vy0 g t

3Examples

Example#1: Pop-Up train Example#2: Shoot the MonkeyExample #3: the football kicker What is the longest kick?

1. We take the vertical motionto find the flight time tf by requesting y(tf) = 0

2. Then we use that time tf to determine the range x(tf) in the horizontal motion.

Example: Projectile Range 1An object is thrown at a speed of 25 m/s. What is the

farthest distance it can travel horizontally before hitting the ground, assuming no air resistance and level ground?

0 y t f vy0 t f 1 2gt f2

R x t f vx0t f

4Example: Projectile Range 2An object is thrown at a speed of 25 m/s. What is the

farthest distance it can travel horizontally before hitting the ground, assuming no air resistance and level ground?

First find tf, for which y becomes zero

0 y t f vy0 t f 1 2gt f2

0 vy0 1 2gt f

t f2vy0

g

Then find out how far the object came in that time

R x t f vx0 t f2vx0 vy0

g

Example: Projectile Range 3An object is thrown at a speed of 25 m/s. What is the

farthest distance it can travel horizontally before hitting the ground, assuming no air resistance and level ground?

Now we have to relate vx0 and vy0 to the initial speed v0 and the angle q

q

Vx0=Vocosq

Vy0=Vosinq

With that, the total range becomes: R2vx0 vy0

g

R2v0

2cos sing

with 2sin cos sin 2

Rv0

2sin 2g

5Example Projectile Range 4The maximal range is achieved at q = 45

Range versus Angle

0

10

20

30

40

50

60

70

0 15 30 45 60 75 90

Angle (degrees)

Ran

ge

(m)

or

Tim

e(s)

TimeRange

Cause and effect.

v In Kinematics we have discussed the motion of objects.

v Now we will talk about why objects move. Dynamics, Forces, Newtons Laws

6Newton's Lawsv Newtons First Law: All objects

continue in their state of rest or of uniform speed in a straight line unless acted on by a net force.

v Newtons Second Law: The net force acting on an object equals its mass times acceleration.

v Newtons Third Law: Whenever an object excerts a force on a second object, the second excerts an equal and opposite force on the first.

F maSir Issac Newton (1642-1727)

==> First we need to talk about Forces !

ForceMost of you intuitively know the effect of forces:v To set an object in motion or change its state of

motionv To deform an object (See also Chapter 9)

In Physics, forces are vectors, with both magnitude and direction

Unit for force : 1 Newton = 1N = 1 kg.m/s2

Forces add up as vectors, the sum of all forces acting on one object is called the net force.

7Stay tuned...

v This Friday: Recitation/CAPA 2v Next Monday: No classes/Labor Day Holiday

v Next Wednesday: Mini-Exam1