1.3.1 Acceleration due to Gravity Defined as: For many years, it was thought that higher mass objects fall towards the Earth more quickly than lower mass objects. This idea was introduced in approximately the year 350 B.C. by Aristotle and was firmly believed until 1590 (nearly 2000 years!). This is when Galileo is thought to have dropped two masses from the leaning tower of Pisa. In fact, many people alive today still think that a heavier mass will fall toward the earth more quickly than a lighter mass. The numerical value of the acceleration that a freely falling object will experience is constant for a given location and is given the symbol. Notice that it is a Galileo Galilei (1564 1642) vector quantity. Since this is another example of uniform acceleration, we can use the kinematics equations we discussed yesterday when solving problems involving free fall. Eg.#1 A diver steps off a 10. m. high diving board with an initial vertical velocity of zero and experiences an average acceleration 2 of 9.8 m/s [down]. Air resistance is negligible. Determine the diver s velocity in metres per second and in kilometres per hour after falling a) 5.00 m. b) 10.m. Page 1 of 10
Eg.#2 An arrow is shot vertically upward beside a building 56 m. high. The initial velocity of the arrow is 37m/s[up]. Air resistance is negligible. At what time does the arrow pass the top of the building on its way up and later, on its way down? Eg.#3 You throw a ball vertically upward and catch it at the height from which you released it. Air resistance is negligible. a) Compare the time the ball takes to rise with the time the ball takes to fall. b) Compare the initial and final velocities. c) What is the instantaneous velocity at the top of the flight? d) What is the ball s acceleration as it is rising? at the top of the flight? as it is falling? e) Sketch the position-time graph, the velocity-time graph, and the accelerationtime graph for the ball s motion during its flight. Use [up] as the positive direction. Page 2 of 10
Eg.#4 Determine the speed at impact in the following situations. 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. b) A steel ball is dropped from the Leaning Tower of Pisa, landing 3.37 s later. Eg.#5 A steel ball is thrown vertically from a ledge so that it has an initial velocity of magnitude 15.0 m/s. The ledge is 15.0 m above the ground. Air resistance is negligible. a) What are the velocity at impact and b) What are the velocity at impact and the total flight time if the initial velocity the total flight time if the initial velocity is upward? is downward? Page 3 of 10
c) Based on your answers to (a) and (b), write a concluding statement. Eg.#6 Show that a free falling mass dropped vertically from rest travels three times 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. Eg.#7 A baseball pitcher throws a ball vertically upward and catches it at the same level 4.2 s later. a) With what initial vertical velocity did the pitcher throw the ball? b) How high does the ball rise? Page 4 of 10
1.3.2 Terminal Velocity Defined as: The magnitude of the terminal velocity depends on two things: #1. 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. Suppose that an elephant and a feather are dropped off a very tall building from the same height at the same time. We will assume the realistic situation that both feather and elephant encounter air resistance. Which object - the elephant or the feather - will hit the ground first? Most people are not surprised by the fact that the elephant strikes the ground before the feather. But why does the elephant fall faster? Is it because the feather experiences more air resistance than the elephant and thus reaches a smaller terminal velocity? If we draw a free-body diagram of the two, we see pretty quickly what causes the elephant to reach the ground first: Page 5 of 10
#2. Another factor that affects terminal velocity is the orientation at which a body falls. If an object falls with a larger surface area perpendicular to the direction of motion it will experience a greater drag force and therefore a smaller terminal velocity. Incidently, a person s terminal speed without a parachute and rolled up into a ball is approximately 89 m/s. With arms and legs extended, the terminal speed is approximately 56 m/s. With a parachute, the terminal speed is approximately 25 m/s. Felix Baumgartner is an Austrian skydiver, daredevil and BASE jumper. He set the world record for skydiving an estimated 39 kilometres, reaching an estimated speed of 1357.64 km/h, or Mach th 1.25, on October 14, 2012, and became the first person to break the sound barrier without vehicular power on his descent. While performing his world record breaking jump, he jumped to Earth from a helium balloon in the stratosphere also setting the altitude record for a manned balloon flight, parachute jump from the highest altitude, and greatest free fall velocity Page 6 of 10
Worksheet 1.3 1. Give an example of freefall where air resistance is negligible Objects falling in a classroom, because of their low speed have negligible air resistance unless their density is quite low 2. A diver entertains tourists in Acapulco, Mexico by diving from a cliff 36 m. above the water. If he starts from rest, determine his landing speed in both m./s. and km./hr. 3. Two high jumpers, one in Java ( ) and the other in London, UK ( ). If they both have an initial velocity of 5.112 m./s.[up], calculate to four significant digits the height that each attains. 4. During the first minutes after liftoff, the space shuttle has an average acceleration of 5g s (i.e. five times the magnitude of the acceleration due to gravity on the surface of the Earth). Calculate the shuttle s speed in metres per second and kilometres per hour after 1.0 m.. Page 7 of 10
5. A person throws a golf ball vertically upward. The ball returns to the same level after 2.6 seconds. a) For how long did the ball rise? b) Determine the initial velocity of the ball c) If the ball were thrown vertically upward on Mars (where ), how long would it remain in flight? 6. During an experiment, a computer determines that the time for a falling steel ball to travel the final 0.80m. before hitting the floor is 0.087s. With what velocity does the ball hit the floor? Page 8 of 10
7. A stone is thrown vertically with a velocity of 14 m./s.[down] from a bridge. a) How long will it take the stone to reach the water 21 m. below? b) What does the negative root in the solution to (a) mean? 8. A steel ball and a tennis ball are dropped simultaneously from a high ledge. The tennis ball encounters significant air resistance and eventually reaches terminal speed. The steel ball essentially undergoes freefall. a) Sketch a velocity time graph a) Sketch a velocity time graph comparing the motions of the two comparing the motions of the two balls taking downward to be positive. balls taking upward to be positive. Page 9 of 10
9. A flowerpot is dropped from the balcony of an apartment, 28.5 m. above the ground. At a time of 1.0 s. after the pot is dropped, a ball is thrown vertically downward from the balcony one story below, 26.0 m. above the ground. The initial velocity of the ball is 12.0m./s.[down] How far above the ground are the two objects when the ball passes the flowerpot? Answers: 2. 3. 4. 5a) b) c) 6. 7a) 9. Page 10 of 10