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One dimensional and When we drop an object it’s continues to increase. That means there is an . Near the earths surface the value of this acceleration is g = 9.8m/s2. This is due to the attractive of gravity. g varies slightly over the earth because the earth is not a perfect and because of the of the earth. On the g is much smaller (~1/6)

For CONSTANT acceleration, The of motion are: 2 v = v0 + at , d = v0t + ½(at ) 2 2 + v = v0 + 2ad , d = ½(v +v0)t (starting at d =0) For this case all quantities are + at is the change in v for interval t And g is rounded up to 10.0 m/s which is 2% off and might fail CHIP Week 2 214 1 Spring 2009 One dimensional motion and gravity

d, vo, v, a, t For CONSTANT acceleration, For constant acceleration, if you The equations of motion are: are told the values of three of these quantities, you can solve for a v = v + at does not involve d 0 fourth quantity --- and not have to 2 solve for the fifth quantity if you d = v0t + ½(at ) does not involve v don’t need to, by picking the proper 2 2 v = v0 + 2ad does not involve t . d = ½(v0 + v)t does not involve a = avg. velocity * time

(starting at d =0) Combine 1st and 4th eqns. Æ 2d eq.

Combine eqs 1 & 4 differently, get 3d eqn., ‘cuz (v-v0)/a = t, plug into eq.4 2 2 d = ½(v0+v)(v-v0)/a = (v -v0 )/2a is same as eqn. 3 Week 2 Physics 214 2 Spring 2009 Throwing vertically In the example shown a ball is thrown vertically.The acceleration is minus + 9.8m/s2 and the motion is symmetric.

v = v0 + at at t=0 v = v0 At the top v = 0 and t = v0/9.8 At the bottom t = 2v /9.8 and v = - v 0 0 g = -9.8m/s2 time up and down = 2 x time to top

2 d = v0t + 1/2at so d=0 when t = 0 or t = -2v0/a or 2v0/9.8 since a = -9.8

2 WHY? d = v0t+1/2at = t(v0+at/2) So d = 0 if t=0 or if ( ) = 0

Week 2 Physics 214 3 Spring 2009 Throwing vertically In the example shown a ball is thrown vertically.The acceleration is minus + 9.8m/s2 and the motion is symmetric.

v = v0 + at at t=0 v = v0 At the top v = 0 and t = v0/9.8 At the bottom t = 2v /9.8 and v = - v 0 0 g = -9.8m/s2 time up and down = 2 x time to top

2 d = v0t + 1/2at so d=0 when t = 0 or t = -2v0/a or 2v0/9.8 since a = -9.8

2 2 v = v0 + 2ad At d = 0 v = +v0 (t = 0) Or v = -v0 ( t = 2v0/9.8)

Week 2 Physics 214 4 Spring 2009 QUIZ Ch 3 E6 Ball is dropped What is the change in velocity between 1 and 4 seconds?

A. 9.8 m/s B. 19.6 m/s v g C. 29.4 m/s D. 39.2 m/s + After 1 sec v = 9.8 t = 9.8 m/s After 4 sec v = 9.8 x 4 = 39.2 m/s Change in velocity is 29.4 m/s .

Week 2 Physics 214 5 Spring 2009 Motion If we throw an object so that it’s initial velocity is horizontal then ignoring it will continue to move with this velocity in the horizontal direction but it will also start to fall so that it’s will be curved.

Week 2 Physics 214 6 Spring 2009 Motion in two dimensions

If we take two axes at right we can analyze the motion along each axis separately and determine properties of the whole motion. We will only deal with cases where there is a constant velocity along one axis and a constant acceleration along the other axis. This means that for up Axis 2 usually y

Axis 1 usually x

down

axis 1 v1 = constant and d1 = v1t note: a1=0 2 2 2 axis 2 v2 = v02 + at d = v02t + ½ at v2 = v02 + 2ad

Week 2 Physics 214 7 Spring 2009 Projectile Motion

We will focus on with a = 9.8m/s2

9.8

v0v θ v1 R = range At the highest point the vertical velocity is zero

vv = v0v + at so t = v0v/9.8 since a = -g 2 h = v0vt + ½ at = vertical height above launch

At impact t = 2v0v/9.8 and R = v1 x 2v0v/9.8 And the vertical velocity is minus v0v o Maximum range R for 45 degree upward launch, easiest to see with trig.

Week 2 Physics 214 8 Spring 2009 Projectile Motion

We will focus on projectiles with a = 9.8m/s2

9.8 v2=vv v0v θ v1 R = range

At impact t = 2v0v/9.8 and R = v1 x 2v0v/9.8

2 2 v1=v0cos(θ) v0v=v0sin(θ) R=2c.s.v0 /9.8m/s 2sin.cos = sin(2θ) , max when 2θ = 90o o Maximum range R for 45 degree upward launch, easiest to see with trig.

Week 2 Physics 214 9 Spring 2009 With no friction there are always two angles which give the same range for the same starting velocity 450 + X and 450 –X

R varies as sin(2θ) http://www.physics.purdue.edu/academic_programs/courses/phys214/movies.php (anim0002.mov)

http://faculty.tcc.fl.edu/scma/carrj/Java/baseball4.html http://www.mhhe.com/physsci/physical/giambattista/proj/projectile.html http://www.physics.purdue.edu/class/applets/phe/projectile.htm

Week 2 Physics 214 10 Spring 2009 Throwing a ball horizontally from a building

Use + down so

v1 g is positive and h is positive 2 g h = v02t + ½ at h v = v02 + at 2 2 v = v02 + 2ah v2

2 R v02 = 0, t = 2h/a

R = v1t = v1sqrt(2h/a)

NOTE h is measured down

Week 2 Physics 214 11 Spring 2009 Summary Chapter 3 Any motion in a plane can analyzed using two axes at right angles. The motion along each axis is independent of the motion along the other axis so the two dimensional motion can be analyzed as two one dimensional linked by time. Special case

axis 1 v1 = constant and d1 = v1t 2 2 2 axis 2 v2 = v02 + at , h = v02t + 1/2at ,v = v02 + 2ah

2 v0v 9.8m/s v1

v1 v1

v For projectile motion a = 9.8m/s2 0v

Week 2 Physics 214 12 Spring 2009 1D-20 Independence of Vertical & Horizontal Motions (Drop-Kick)

One ball drops from rest. The other ball is simultaneously projected horizontally

Which ball will hit the ground first ?

Listen to the SOUND when they hit the ground and when they bounce.

THE VERTICAL & HORIZONTAL MOTIONS ARE INDEPENDENT. THE HORIZONTAL VELOCITY DOES NOT AFFECT THE VERTICAL MOTION. THE VERTICAL FALL TIME IS THE SAME AS LONG AS THE BALLS DROP SIMULTANEOUSLY FROM THE SAME HEIGHT.

Week 2 Physics 214 13 Spring 2009 1D-21 Independence of Vertical and Horizontal Motions

A ball is projected vertically from a cart traveling horizontally

The trajectory in the cart frame The trajectory in the room frame

THE HORIZONTAL MOTION OF THE BALL IS UNAFFECTED BY ITS VERTICAL MOTION.

Week 2 Physics 214 14 Spring 2009 1G-03 Measurement of g

Measuring g by dropping an object

d = ½ gt2 t = sqrt ( 2d/g ) g = 2d/t2

Let d = 1 m, What difficulties might be t = sqrt(2/9.8) = 0.45 s encountered in measuring h & t of fall ?

Hand timing would not be accurate because of the short fall time

Week 2 Physics 214 15 Spring 2009 1K-11 Coin and Feather

DROPPING A COIN AND A FEATHER ?

DO ALL OBJECTS HAVE THE SAME ACCELERATION WHEN DROPPED ?

IN AIR, WEIGHT AND SURFACE AREA MAKE OBJECTS FALL AT DIFFERENT BECAUSE OF AIR FRICTION.

Week 2 Physics 214 16 Spring 2009 1D-22 Water Jets & Projectile Motion

PROJECTILE MOTION OF A WATER JET What gives the maximum range?

g

NEGLECTING FRICTION THE RANGE IS A MAXIMUM AT 450. TWO DIFFERENT ANGLES CAN GIVE THE SAME RANGE (ANGLES SYMMETRIC ABOUT 45°). A LARGER ANGLE MEANS A LONGER TIME OF FLIGHT, BUT LESS HORIZONTAL VELOCITY. A SMALLER ANGLE MEANS A LARGER HORIZONTAL VELOCITY, AND LESS FLIGHT TIME. THE TRAJECTORY IS SYMMETRIC.

Week 2 Physics 214 17 Spring 2009 1D-23 Shoot the Monkey

The monkey falls out of the tree at the instant the gun is fired

v0y/v0x = h/d WHERE SHOULD ONE AIM, ABOVE, BELOW OR AT? y=0 .

|------d------| Turn off g, bullet hits monkey at height h. Turn on g, both bullet and monkey fall same amount 2 Ignoring friction y = v0yt – 1/2gt , t = d/v0x , v0yt = v0yd/v0x = h So at x = d y = h – ½ gt2 In the same time the monkey falls ½ gt2

So the bullet always hits the monkey no matter what the value of v0, Unless range R is less than d! THE VERTICAL MOTION IS INDEPENDENT OF THE HORIZONTAL MOTION THE EFFECT OF FRICTION IS MINIMIZED BY USING A HEAVY TARGET Week 2 Physics 214 18 Spring 2009 Questions Chapter 3

Q1 A small piece of paper is dropped and flutters to the floor. Is the piece of paper accelerating at any time during this motion? Explain?

Yes at the start

Q4 A lead ball and an aluminum ball, each 1 in. in diameter, are released simultaneously and allowed to fall to the ground. Due to its greater density, the lead ball has a substantially larger than the aluminum ball. Which of these balls, if either, has the greater acceleration due to gravity? Explain. They both have the same gravitational acceleration. Any difference in how they fall is due to friction. If the balls have identical shapes size and surface polish the friction is the same but the weight is different, so the two motions are .not quite identical Week 2 Physics 214 19 Spring 2009 QUIZZES

Q8 A rock is dropped from the top of a diving platform into the swimming pool below. Will the traveled by the rock in a 0.1-second interval near the top of its flight be the same as the distance covered in a 0.1-second interval just before it hits the water? Explain. . No because the velocity is increasing

Q10 A ball is thrown downward with a large starting velocity. A. Will this ball reach the ground sooner than one that is dropped from rest at the same time from the same height? Explain. B. Will this ball accelerate more rapidly than one that is dropped with no initial velocity? Explain. . Yes because it will have a higher average velocity

. No the acceleration is the same

Week 2 Physics 214 20 Spring 2009 Q14 A ball is thrown straight upward. At the very top of its flight, the velocity of the ball is zero. Is its acceleration at this point also zero? Explain. No the acceleration is 9.8m/s2 down

Q15 A ball rolls up an inclined plane, slows to a stop, and then rolls back down. Do you expect the acceleration to be constant during this process? Is the velocity constant?

The acceleration is constant the velocity is not

Q19 Is it possible for an object to have a horizontal component of velocity that is constant at the same time that the object is accelerating in the vertical direction? Explain by giving an example, if possible. Yes a projectile

Week 2 Physics 214 21 Spring 2009 Ch 3 E4

Heart beat = 75 beats/minute a) What is the time between pulses? b) How far does an object fall in this time? (from rest)

60 a) t = /75 = 0.8 s

9.8m/s2 2 b) d = v0t + ½ 9.8t = 3.136 m

Week 2 Physics 214 22 Spring 2009 Ch 3 E8 t = 1.53 s Ball thrown up at 15 m/s a) How high after 1 second? g b) How high after 2 seconds? t = 2 s + 15 m/s

2 After 1 sec d = v0t + ½ at = 15 – 4.9 = 10.1 m After 2 sec d = 15 x 2 – ½ 9.8 x 22 = 10.4 m 15 Time to top v = v0 + at t = /9.8 = 1.53 s Height at top d = 11.48m

Week 2 Physics 214 23 Spring 2009 Ch 3 E10

2 V0 = 18 m/s a = - 2 m/s a) What is v after 4 seconds? a=2m/s2 b) What is time to top? + 18 m/s

a) v = v0 + at = 18 – 2 x 4 = 10m/s

18 b) v = 0 t = /2 = 9s

Week 2 Physics 214 24 Spring 2009 Ch 3 E16

g V0v = 30 m/s V0H = 30 m/s g = - 9.8m/s2 30 m/s a) What is time to top? b) What is the range? + 30 m/s

30 a) v = v0 + at t = /9.8 = 3.06s tR = 6.12s

b) d = 30 x tR = 183.6m

Week 2 Physics 214 25 Spring 2009 Ch 3 CP2 V01 = 0 m/s V02 = 12 m/s a) What are the after 1.5s? b) How far has each ball dropped in 1.5s? 1 2 c) Does the velocity difference change? 12 m/s

a) v1 = at = 9.8 x 1.5 = 14.7m/s

v2 = 12 + 9.8 x 1.5 = 26.7m/s 2 b) d1 = ½at = 11.03m 2 d2 = v2t + ½at = 29.03m c) No

Week 2 Physics 214 26 Spring 2009 Ch 3 CP4

V0v = 200m/s v0H = 346m/s g a) How long in the air? 200m/s b) How far? 346m/s c) v0v = 346 v0H = 200

200 a) v = v0 + at time to top = /9.8 = 20.4s 400 time to range = /9.8 = 40.8s b) d = 346 x 40.8 = 14120m

692 c) ↑346 →200 time = /9.8 = 70.6s d = 200 x 70.6 = 14120

Week 2 Physics 214 27 Spring 2009 Extinction of the dinosaurs There are accurate ways to look at events in geologic time back to the formation of the earth about 4.5 billion years ago. ™Radioactive elements are very accurate clocks and sedimentary layers reveal geologic events as a of time. ™65 million years ago the extinction of the dinosaurs ™250 million years ago over 90% of all species became extinct. Experimental measurements show that the 65 million extinction occurred about when a large asteroid impacted in the gulf of Mexico near the Yucatan peninsula, enveloping the Earth in a cloud of dust.

Week 2 Physics 214 28 Spring 2009 Sedimentary layers Looking back in time

Iridium Hill Montana

Week 2 Physics 214 29 Spring 2009 Sedimentary layers

Week 2 Physics 214 30 Spring 2009 Asteroid impact simulation

Less than a minute after impact, the dissipation of the asteroid kinetic produces a stupendous explosion that melts, vaporizes, and ejects a substantial volume of calcite, granite, and water. The dominant feature here is the conical curtain of hot debris that has been ejected and is now falling back to Earth. The turbulent material inside this curtain is still being accelerated by the explosion from the crater excavation.

Week 2 Physics 214 31 Spring 2009