This Week Gravity Dropping and throwing objects From

This Week • Gravity: Dropping and throwing objects From Baseballs to satellites • What causes motion. All changes involve motion • Mass and Weight What’s the difference? • Friction good and bad A necessary evil. • Extinction of the dinosaurs 12/6/2020 Physics 214 Summer 2017 1

One dimensional motion and gravity When we drop an object it’s velocity continues to increase that means there is an acceleration. Near the earths surface the value of this acceleration is g = 9. 8 m/s 2. This is due to the attractive force of gravity and g does vary over the earth because the earth is not a perfect sphere and because of the rotation of the earth. On the moon g is much smaller The equations of motion are: v = v 0 + at d = v 0 t + 1/2 at 2 v 2 = v 02 + 2 ad d = ½(v + v 0)t (starting at d =0) 12/6/2020 + For this case all quantities are + Physics 214 Summer 2017 2

Throwing vertically In the example shown a ball is thrown vertically. The acceleration is minus 9. 8 m/s 2 and the motion is symmetric. v = v 0 + at at t=0 v = v 0 At the top v = 0 and t = v 0/9. 8 At the bottom t = 2 v 0/9. 8 and v = - v 0 + g = -9. 8 m/s 2 time up and down = 2 x time to top d = v 0 t + 1/2 at 2 so d=0 when t = 0 or t = -2 v 0/a or 2 v 0/9. 8 since a = -9. 8 v 2 = v 02 + 2 ad At d = 0 v = +v 0 (t = 0) Or v = -v 0 ( t = 2 v 0/9. 8) 12/6/2020 Physics 214 Summer 2017 3

Projectile Motion If we throw an object so that it’s initial velocity is horizontal then ignoring friction it will continue to move with this velocity in the horizontal direction but it will also start to fall so that it’s trajectory will be curved. 12/6/2020 Physics 214 Summer 2017 4

Motion in two dimensions If we take two axes at right angles 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 v 1 = constant and d 1 = v 1 t axis 2 v 2 = v 02 + at d = ½(v 2 + v 02)t 12/6/2020 d = v 02 t + 1/2 at 2 v 22 = v 022 + 2 ad Physics 214 Summer 2017 5

Projectile Motion We will focus on projectiles with a = 9. 8 m/s 2 9. 8 v 02 v 1 R At the highest point the vertical velocity is zero v 2 = v 02 + at so t = v 02/9. 8 h = v 02 t + 1/2 at 2 At the end t = 2 v 02/9. 8 and R = v 1 x 2 v 02/9. 8 And the vertical velocity is minus v 02 12/6/2020 Physics 214 Summer 2017 6

1 K-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 SPEEDS BECAUSE OF AIR FRICTION. 12/6/2020 Physics 214 Summer 2017 7

1 G-05 Measurement of g Measuring g by dropping an object d = 1/2 gt 2 t = sqrt ( 2 d/g ) g = 2 d/t 2 What difficulties might be encountered in measuring h & t of fall ? A steel ball is held then released by an electromagnet. Photo electric sensors are connected to timers. The top display shows the time between sensor 1 (the release point) and sensor 2 (the middle sensor). The bottom display shows the time between sensor 1 and 3 (at the bottom). Hand timing would not be accurate because of the short fall time 12/6/2020 Physics 214 Summer 2017 8

1 D-22 Water Jets & Projectile Motion PROJECTILE MOTION OF A WATER JET What angle 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. 12/6/2020 Physics 214 Summer 2017 9

Trajectories With no friction there always two angles which give the same range for the same starting velocity 450 + X and 450 - X anim 0002. mov http: //faculty. tcc. fl. edu/scma/carrj/Java/baseball 4. html http: //www. mhhe. com/physsci/physical/giambattista/projectile. html http: //www. physics. purdue. edu/class/applets/phe/projectile. htm 12/6/2020 Physics 214 Summer 2017 10

Throwing a ball horizontally from a building v 1 g h v R Use + down so g is positive and h is positive h = v 02 t + 1/2 at 2 v 2 = v 02 + at v 22 = v 022 + 2 ah v 02 = 0, t 2 = 2 h/a R = v 1 t NOTE h is measured down 12/6/2020 Physics 214 Summer 2017 11

Summary Chapter 3 v. Any motion in a plane can analyzed using two axes at right angles. v. The motion along each axis is independent of the other v. The two dimensional motion can be analyzed as two one dimensional motions linked by time. Special case axis 1 v 1 = constant and d 1 = v 1 t axis 2 v 2 = v 02 + at , h = v 02 t + 1/2 at 2 , v 22 = v 022 + 2 ah At the end t = 2 v 02/9. 8 and R = v 1 x 2 v 02/9. 8 v 02 v 1 9. 8 m/s 2 v 1 R 12/6/2020 Physics 214 Summer 2017 v 02 12

1 D-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. 12/6/2020 Physics 214 Summer 2017 13

1 D-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. 12/6/2020 Physics 214 Summer 2017 14

1 D-23 Shoot the Monkey The monkey falls out of the tree at the instant the gun is fired WHERE SHOULD ONE AIM, ABOVE, BELOW OR AT? Ignoring friction y = v 0 yt – 1/2 gt 2 t = x/v 0 x , v 0 y/v 0 x = h/d at x = d y = h – 1/2 gt 2 In the same time the monkey falls 1/2 gt 2 So the bullet always hits the monkey no matter what the value of v 0 THE VERTICAL MOTION IS INDEPENDENT OF THE HORIZONTAL MOTION THE EFFECT OF FRICTION IS MINIMIZED BY USING A LARGE TARGET 12/6/2020 Physics 214 Summer 2017 15

Chapter 4. What causes motion In our everyday life we observe that objects change their state of motion. In fact everything that happens in the Universe results from a change in motion. That is a static inert object does not contribute to any of the things we consider to be useful. The functioning of our body depends on continual change throughout our bodies. These changes are produced by forces and in our everyday life there are just two forces. Gravity acts on mass F = Gm 1 m 2/r 2 Electric charge F = kq 1 q 2/r 2 12/6/2020 Physics 214 Summer 2017 16

Newtons Second and First Law Second Law The acceleration of an object is directly proportional to the magnitude of the imposed force and inversely proportional to the mass. The acceleration is in the same direction as the force F = ma F and a are vectors unit is a Newton (or pound) 1 lb = 4. 448 N First Law An object remains at rest or in uniform motion in a straight line unless it is acted on by an external force. F = 0 a = 0 so v = constant anim 0003. mov anim 0004. mov 12/6/2020 Physics 214 Summer 2017 17

Mass and weight Newtons second law enables us to measure relative mass. If we apply the same force to two objects and measure the accelerations then. F = m 1 a 1 and F = m 2 a 2 so m 1/m 2 = a 2/a 1 We then need to have one mass as a calibration and a kilogram is the mass of a piece of platinum held in Paris. Since gravity acts proportional to mass then the force near the earths surface is F = mg this is the weight of an object so if we compare F 1 = m 1 g and F 2 = m 2 g then weight 1/weight 2 = m 1/m 2 12/6/2020 Physics 214 Summer 2017 18

Inertia = tendency of an object to resist changes in its velocity. Since F = ma and a = Δv/t then Ft = mΔv So if a force acts for a time t the change in velocity will be smaller for larger masses so it is mass that determines inertia. In particular if t is very small and m is large then F can also be large but Δv can still be very small. 12/6/2020 Physics 214 Summer 2017 19

1 F-04 Brass Rod (Inertia) How to remove the paper without toppling the rod ? One needs to remove the paper quickly so that the frictional force only lasts for a short time and the inertia of the rod prevents it from toppling over. 12/6/2020 Physics 214 Summer 2017 20

1 F-05 Coin, Hoop & Milk Bottle (Inertia) How can you get the coin into the bottle without touching it ? This is actually a trick which depends on hitting the ring so that the top deflects down and the coin is free to drop 12/6/2020 Physics 214 Summer 2017 21

1 F-06 Inertial Ball Which string breaks first ? Case 1: Place the aluminum rod in the lower loop and pull SLOWLY downward. Case 2: Use the wooden mallet to strike a sharp blow to the aluminum rod. IF IT IS DONE SLOWLY THE UPPER STRING BREAKS FIRST BECAUSE THE TENSION IN THAT STRING WILL BE THE WEIGHT OF THE BALL PLUS THE TENSION IN THE LOWER STRING. IF THE LOWER STRING IS STRETCHED SUFFICIENTLY RAPIDLY, IT WILL REACH ITS BREAKING POINT BEFORE THE BALL HAS A CHANCE TO MOVE APPRECIABLY. 12/6/2020 Physics 214 Summer 2017 22

Friction In our everyday world any object which is moving feels a force opposing the motion --- this is friction. v An object which is sliding v. The air resistance on your car These types of friction result in energy being lost and minimizing friction is very important. But Friction is also useful and essential since with no friction va car would not move but just spin it’s wheels va car would not be able to turn a corner vwe would not be able to walk vobjects would slide off surfaces unless perfectly horizontal 12/6/2020 Physics 214 Summer 2017 23

Force diagrams + When we are analyzing a particular object we have to take into account all the forces acting on the body both in magnitude and direction. The acceleration of the object is equal to F/m in the direction of F where F is the net force acting. As in the example above we know that there is a force called friction which opposes the motion. In the case shown the 10 N force is + and the 2 N force so the net force is 8 N and 8 = 5 a 12/6/2020 Physics 214 Summer 2017 24

Our World 12/6/2020 Physics 214 Summer 2017 25

Extinction of the dinosaurs There accurate ways to look at events in geologic time back to the formation of the earth about 4. 5 billion years ago. v. Radioactive elements are very accurate clocks and sedimentary layers reveal geologic events as a function of time. v 65 million years ago the extinction of the dinosaurs and 70% of all species v 250 million years ago over 90% of all species became extinct. Experimental measurements show that the 65 million event was due to an asteroid which impacted in the gulf of Mexico near the Yucatan peninsula. An asteroid about 6 miles in diameter and traveling at 45, 000 miles a second, slammed into the Gulf of Mexico causing a crater 24 miles deep and 125 miles wide. The blast was equivalent to 100 million tons of TNT. 12/6/2020 Physics 214 Summer 2017 26

Sedimentary layers Looking back in time Iridium Hill Montana 12/6/2020 Physics 214 Summer 2017 27

Sedimentary layers 12/6/2020 Physics 214 Summer 2017 28

Asteroid impact simulation Less than a minute after impact, the dissipation of the asteroid kinetic energy 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. 12/6/2020 Physics 214 Summer 2017 29

Worked Questions and Problems 12/6/2020 Physics 214 Summer 2017 30

Questions Chapter 3 Q 1 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 Q 4 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 mass 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 so if the balls have identical shapes , size and surface polish the two motions will be identical 12/6/2020 Physics 214 Summer 2017 31

Q 8 A rock is dropped from the top of a diving platform into the swimming pool below. Will the distance 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 Q 10 A ball is thrown downward with a large starting velocity. A. Will this ball reach the ground sooner than one that is just dropped 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 12/6/2020 Physics 214 Summer 2017 32

Q 14 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. 8 m/s 2 down Q 15 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 Q 19 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 12/6/2020 Physics 214 Summer 2017 33

Ch 3 E 4 Heart beat = 75 beats/minute a) What is the time between pulses? b) How far does an object fall in this time? a) t = 60/75 = 0. 8 s b) d = v 0 t + ½ 9. 8 t 2 = 3. 136 m 12/6/2020 Physics 214 Summer 2017 9. 8 m/s 2 34

Ch 3 E 6 Ball is dropped What is the change in velocity between 1 and 4 seconds? v g 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 12/6/2020 Physics 214 Summer 2017 35

Ch 3 E 8 Ball thrown up at 15 m/s a) How high after 1 second? b) How high after 2 seconds? t = 1. 53 s g + t=2 s 15 m/s After 1 sec d = v 0 t + ½ at 2 = 15 – 4. 9 = 10. 1 m After 2 sec d = 15 x 2 – ½ 9. 8 x 22 = 10. 4 m Time to top v = v 0 + at Height at top 12/6/2020 t = 15/9. 8 = 1. 53 s d = 11. 48 m Physics 214 Summer 2017 36

Ch 3 E 10 V 0 = 18 m/s a = - 2 m/s 2 a) What is v after 4 seconds? b) What is time to top? a=2 m/s 2 + 18 m/s a) v = v 0 + at = 18 – 2 x 4 = 10 m/s b) v = 0 12/6/2020 t = 18/2 = 9 s Physics 214 Summer 2017 37

Ch 3 E 16 V 0 v = 30 m/s V 0 H = 30 m/s g = - 9. 8 m/s 2 a) What is time to top? b) What is the range? a) v = v 0 + at g 30 m/s + 30 m/s t = 30/9. 8 = 3. 06 s t. R = 6. 12 s b) d = 30 x t. R = 183. 6 m 12/6/2020 Physics 214 Summer 2017 38

Ch 3 CP 2 V 01 = 0 m/s V 02 = 12 m/s a) What are the velocities after 1. 5 s? b) How far has each ball dropped in 1. 5 s? c) Does the velocity difference change? 1 2 12 m/s a) v 1 = at = 9. 8 x 1. 5 = 14. 7 m/s v 2 = 12 + 9. 8 x 1. 5 = 26. 7 m/s b) d 1 = ½at 2 = 11. 03 m d 2 = v 2 t + ½at 2 = 29. 03 m c) No 12/6/2020 Physics 214 Summer 2017 39

Ch 3 CP 4 V 0 v = 200 m/s v 0 H = 346 m/s a) How long in the air? b) How far? c) v 0 v = 346 v 0 H = 200 a) v = v 0 + at g 200 m/s 346 m/s time to top = 200/9. 8 = 20. 4 s time to range = 400/9. 8 = 40. 8 s b) d = 346 x 40. 8 = 14120 m c) ↑ 346 → 200 time = 692/9. 8 = 70. 6 s d = 200 x 70. 6 = 14120 12/6/2020 Physics 214 Summer 2017 40