Unit #1: Dynamics. Introduction. Example #

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1 Unit #1: Dynamics Lesson #4: Accelera4on due to Gravity 1/24 Introduction Accelera4on due to gravity is defined as the accelera4on of an object caused by the force of gravity. The accelera4on due to gravity of any object in freefall is the same (ignoring air resistance) because it doesn t depend on the object s mass. Accelera4on due to gravity is given the symbol g On the surface of Earth, the accelera4on due to gravity is approximately g =9.8 m. s. 2 [down]. 2/24 Example #1 Eg.#1 A diver steps off a 10. m. high diving board with an ini4al ver4cal velocity of zero and experiences an average accelera4on of 9.8 m/s 2 [down]. Determine the diver s velocity in metres per second and in kilometres per hour aver falling 5.00 m. v f 2 = v i 2 +2a d v f = 2a d = m. s m =9.9 m. s. The final velocity of the diver would be 9.9m./s.[down] 3/24 1

2 Example #2 Eg. #2. An arrow is shot ver4cally upward at 37 m./s.[up] beside a 56 m. high building. Air resistance is negligible. How long does it take the arrow to pass the top of the building on its way up and later, on its way down? d = v i t+ 1/2 a t 2 56m.[up]=37 m. s. [up] t 1/2 9.8 m. s. 2 t 2 let x= t 4.9 x 2 37x+56=0 Solving using the quadra4c formula gives t=2.1s, 5.5s 4/24 Example #3 Eg. #3. You throw a ball ver4cally upward and catch it at the height from which you released it. Air resistance is negligible. a) Compare the 4me the ball takes to rise with the 4me the ball takes to fall. They are the same b) Compare the ini4al and final veloci4es. They are equal and opposite 5/24 Example #3 cont d c) What is the instantaneous velocity at the top of the flight? Zero d) What is the ball s accelera4on as it is rising? at the top of the flight? as it is falling? 9.8 m/s 2. 6/24 2

3 Example #3 cont d e) Sketch the posi4on- 4me graph, the velocity- 4me graph, and the accelera4on- 4me graph for the ball s mo4on during its flight. Use [up] as the posi4ve direc4on. 7/24 Example #4 Eg. #4. Determine the speed at impact in the following situa4ons. Air resistance is negligible. a) A ring- billed gull drops a shellfish onto a rocky shore from a height of 12.5 m to crack the shell. v f 2 = v i 2 +2a d v f = 2a d = m. s m =16 m. s. The final velocity of the shellfish would be 16m./s.[down] 8/24 Example #4 cont d b) A steel ball is dropped from the Leaning Tower of Pisa, landing 3.37 s later. v f = v i + a t =0+9.8 m. s. 2 [down] 3.37s. =33 m. s. [down] 9/24 3

4 Example #5 Eg.#5 A steel ball is thrown ver4cally from a ledge so that it has an ini4al velocity of magnitude 15.0 m/s. The ledge is 15.0 m above the ground. Air resistance is negligible. a) What is the velocity at impact if the ini4al velocity is upward? v f 2 = v i 2 +2a d v f = v i 2 +2a d = (15.0 m. s. ) m. s m =23 m. s. 10/24 The final velocity is 23m./s.[down] Example #5 cont d b) What is the total flight 4me if the ini4al velocity is upward? t= v f v i / a av = 23 m. s. [down] 15.0 m. s. [up]/9.52 m. s. 2 [down] = 23 m. s. [down]+15.0 m. s. [down]/9.52 m. s. 2 [down] =3.9s. The flight 4me is 3.9s. 11/24 Example #5 cont d c) What is the velocity at impact if the ini4al velocity is downward? v f 2 = v i 2 +2a d v f = v i 2 +2a d = (15.0 m. s. ) m. s m =23 m. s. The final velocity is 23m./s.[down] 12/24 4

5 Example #5 cont d d) What is the total flight 4me if the ini4al velocity is downward? t= v f v i / a av = 23 m. s. [down] 15.0 m. s. [down]/9.52 m. s. 2 [down] =0.80s. The flight 4me is 0.80s. 13/24 Example #5 cont d e) Based on your answers to (a) through (d), write a concluding statement. The ini4al direc4on of the ini4al velocity made no difference to the final velocity. The total flight 4me was affected by the direc4on of the ini4al velocity. 14/24 Example #6 Eg.#6 Show that a free falling mass dropped ver4cally from rest travels three 4mes as far from t = 1.0 s to t = 2.0 s as it does from t = 0.0 s to t = 1.0 s. d 0 1 = v i t+ 1/2 a t 2 d 0 2 = v i t+ 1/2 a t 2 =0+ 1/2 9.8 m. s. 2 [down] (1.0s.) 2 =0+ 1/2 9.8 m. s. 2 [down] (2.0s.) 2 =4.9m.[down] =20.m.[down] d 1 2 =20.m.[down] 4.9m.[down] =15m. 15/24 5

6 Example #7 Eg.#7 A baseball pitcher throws a ball ver4cally upward and catches it at the same level 4.2 s later. a) With what ini4al ver4cal velocity did the pitcher throw the ball? v i = v f a av t =0 9.8 m. s. 2 [down] 2.1s. = 21 m. s. [down] =21 m. s. [up] The pitcher threw the ball with a velocity of 21 m./s.[up]. 16/24 Example #7 b) How high does the ball rise? d =( v i + v f /2 ) t =( 21 m. s. [up]+0/2 ) 2.1s. =21m.[up] The ball rose 21m.[up]. 17/24 Terminal Velocity Defined as: the maximum velocity aiained by a falling object. The magnitude of the terminal velocity depends on two things: #1. The weight of the falling object. #2. The cross sec4onal area of the object perpendicular to the direc4on of mo4on. 18/24 6

7 Terminal Velocity and Weight Heavier objects usually have a greater terminal velocity. For an object to experience terminal velocity, air resistance must balance weight. This happens more quickly for lighter objects than for heavier objects. Thus heavier objects accelerate longer and experience a greater terminal velocity. 19/24 Terminal Velocity and Weight Suppose that an elephant and a feather are dropped off a very tall building from the same height at the same 4me. We will assume the realis4c situa4on that both feather and elephant encounter air resistance. Which object - the elephant or the feather - will hit the ground first? 20/24 Terminal Velocity and Weight Most people are not surprised by the fact that the elephant strikes the ground before the feather. The reason this happens of course is because the feather reaches terminal velocity quickly and no longer accelerates whereas the elephant con4nues to accelerate for a much longer 4me. 21/24 7

8 Terminal Velocity and Area If an object falls with a larger surface area perpendicular to the direc4on of mo4on it will experience a greater drag force and therefore a smaller terminal velocity. To illustrate this point consider the terminal velocity of a person: without a parachute and rolled up into a ball: 89 m/s. with arms and legs extended: 56 m/s. with a parachute: 25 m/s. 22/24 Felix Baumgartner Felix Baumgartner is an Austrian skydiver, daredevil and BASE jumper. He set the world record for skydiving an es4mated 39 kilometres, reaching an es4mated speed of km/h, or Mach 1.25, on October 14th, 2012, and became the first person to break the sound barrier without vehicular power on his descent. 23/24 Felix Baumgartner While performing his world record breaking jump, he jumped to Earth from a helium balloon in the stratosphere also semng the al4tude record for a manned balloon flight, parachute jump from the highest al4tude, and greatest free fall velocity 24/24 8