Scalar versus Vector Quantities. Motion: Kinematics. Direction Matters. In the Universe. Speed and Velocity. Distance vs.

Transcription

1 Scalar versus Vector Quantities Scalar Quantities Magnitude (size) 55 mph Motion: Kinematics Chapter 2 Vector Quantities Magnitude (size) Direction 55 mph, North Example One Dx = +2 m Example Two Direction Matters 1 m 3 m In the Universe Two Possible Situations for Everything: 1. Constant velocity 2. Acceleration speeded up or slowed down (requires a force) What about an object that is just sitting there? 1 m 4 m Dx = -3 m Distance vs. Displacement Distance = total length of travel Speed and Velocity Ave Speed = distance (in meters) time (in seconds) Displacement = x f - x i = Dx (from home) Ave Velocity = displacement = time Dx Dt 1

2 An athlete sprints for 50.0 m in 8.00 s. She then stops and slowly walks back to the starting line in 30.0 s. a. Calculate the average speed b. Calculate the average velocity c. Suppose she only walked halfway back to the starting line in 15.0 s. i. Calculate the average speed. ii. Calculate her average velocity. During a 3.00 second time interval, a runner s position changes from x 1 = 50.0 m to x 2 =30.5 m. What was her average velocity? (Ans: m/s) A plane flies miles east in 2.00 hours. The plane then turns around and flies west for miles in 1.00 hour. a) Calculate the average speed (133 mph) b) Calculate the average velocity. (67.0 mph, east) A bicycle has an average speed of 5miles/hr. How far did it go in two hours? How long would it take a train traveling 72 km/hr to travel 60 km? A car is driven at an average speed of km/hr for two hours, then driven at an average speed of 50.0 km/hr for the next hour. What was the average speed for the three-hour trip? (Ans: 83.3 km/hr) 2

3 Since the speed changed, we have to address each part of the trip separately. 1. To find the distance for the first two hours, we use: x=vt x= 100km/hr X 2 hr = 200 km 3. To find the average speed, we will divide the total distance by the time: v=x/t = 250 km /3 hr = 83.3 km/hr 2. To find the distance for the last hour, we use the same process: x= 50km/hr X 1 hr = 50 km Average vs. Instantaneous Speed Average Speed averaged for an entire trip Instantaneous Speed Speed at an infinitely short interval Constant velocity Changing velocity Two dog owners are 10.0 m apart. They walk towards each other at 1.20 m/s. In the meantime, their dog runs back and forth between the two owners at 2.50 m/s. How far will the dog have travelled when they meet in the middle? Acceleration Acceleration The rate of change of velocity 1. Speed (speeding up or slowing down) Speeding up or slowing down 2. Direction (turning a bicycle) 3

4 a = Dv = v final v initial Dt t final t initial What is the acceleration of a bicycle that moves from rest to 5 m/s in 3 seconds? Units = m/s 2 Acceleration can be negative (slowing down, decelerating) A spaceship is flying through space and slows down from 24,000 m/s to 20,000 m/s in 500 seconds. What is the acceleration? What is unusual about this number? A car starts from a complete stop at a stoplight and accelerates at 5 m/s 2 for 10 seconds. What is its final velocity? (Ans = 50 m/s) A car moving at 30 m/s decelerates at 3.0 m/s 2. How long will it take for it to come to a complete stop? Kinematic Equations (Constant acceleration only) a = v f v i t x = x i + v i t + ½ at 2 a = v f 2 - v i 2 2x v = v f + v i 2 (ANS: t = 10 seconds) 4

5 An airplane takes off at 27.8 m/s and can accelerate at 2.00 m/s 2. How long must the runway be if the plane is to take off safely? How long does it take a car to cross a 30.0 m intersection, from rest, if it accelerates at 2.00 m/s 2? (Ans: 193 m) (Ans: 5.48 s) Warm Up Determine the final velocity and time for the Hot Wheels car as it goes down the track. Calculate acceleration. Calculate how far the car would travel on the level in 10 seconds. Would it really travel this far? A pitcher throws a baseball at 44 m/s. What is the acceleration if the ball travels through 3.5 m from the start of the pitch to the release? (Ans: 280 m/s 2 ) You wish to design an airbag that can protect a driver from a head-on collision at km/h. Assume the car crumples on impact a distance of 1.00 m. a) What is the driver s acceleration? (-386 m/s 2 ) b) How fast does the airbag need to act? (0.072 s) A driver is moving at 28 m/s. It takes him 0.50 seconds to react and slam on the brakes to avoid hitting a sasquatch. At that time, the brakes provide an acceleration of 6.0 m/s 2. Calculate the total stopping distance. (Ans: 79 m) 5

6 A horse starts from rest and can reach a speed of 4.5 m/s in 5 seconds. a) Calculate the average speed of the horse. (2.25 m/s) b) Calculate the acceleration of the horse. (0.9 m/s 2 ) c) Calculate the distance the horse will travel in 35 seconds, assuming the horse keeps accelerating. (551 m) Ben Movin is waiting at a stoplight. When it finally turns green, Ben accelerated from rest at a rate of 6.00 m/s 2 for a time of 4.10 seconds. a. Calculate the displacement of Ben's car during this time period. b. Calculate his final speed. c. Calculate the average speed during this period. Falling Objects Aristotle (Greece, ~400 B.C.) Heavier object fall faster than lighter ones Aristotle versus Galileo. In the steel cage. Galileo (1600 s, Renaissance) All objects fall at the same rate (near the surface of the Earth, in a vacuum) Graph Galileo s Numbers Galileo found that an object is at the following heights over time (from the Leaning Tower) Time(s) Height(m)

7 The displacement gets greater as it falls Acceleration due to Gravity Can act at a distance, is constant near the surface of the earth g = 9.80 m/s 2 Usually set the top of the drop at zero Gravity Why don t people fall off the earth at the South Pole? y o = 0 Do things accelerate forever? Not in an atmosphere. Air drag causes terminal velocity Otherwise raindrops would hit us with deadly speed. Terminal velocity Fastest a human can drop in air 120 miles/hr (about 60 m/s). ~150 m Gravity: Freefall Freefall falling only under the acceleration of gravity Object was dropped, not thrown down y = y i + v i t + ½ at 2 Simplifies to: y = ½ at 2 7

8 A rock is dropped from a cliff, calculate how far it falls during the following times. Also, draw a sketch of how it falls. Time Distance 1s 2s 3s 4s 5s A rock is dropped from a cliff, calculate how far it falls during the following times. Also, draw a sketch of how it falls. Time Distance 1s -4.9 m 2s m 3s m 4s m 5s -123 m 0 s 0 m 1 s -4.9 m A rock is dropped into a well and hits the water in 2.45 seconds. How deep is the well? 2 s m 3 s m 4 s m (Ans: 29.4 m) At what speed did the rock hit the bottom of the well? A ball is thrown off a cliff with a initial velocity of m/s. Calculate its position and speed after 1.00 s and 2.00 s. (Ans: 24.0 m/s) (Ans: m, m/s & m, m/s) 8

9 A person throws a ball into the air with an initial velocity of 15.0 m/s. a) How high will the ball go? (Hint: use 9.80 m/s 2 ) (Ans: 11.5 m) b) How long is it in the air? (Ans: 3.06 s) c) What is the ball s velocity when it returns to the boy s hand? (Ans: m/s) d) Does the ball have zero acceleration at its highest point? What time will the ball pass a point 8.00 m above the person s hand? y = y i +v i t + ½at 2 8 = 0 +15t +1/2(-9.8)t 2 8 = 15t -4.9t 2 0 = 4.9t 2-15t +8 a = 4.9 b = -15 c = 8 Ans: 0.69 s and 2.37 s The Lava Blob A volcano shoots a blob of lava upward at 23.2 m/s. a. Calculate the maximum height. (27.5 m) b. Calculate the time to reach the maximum height. (2.37 s) c. Calculate the total time in the air. (4.73 s) Arrow example An arrow is shot into the air and reaches a maximum height of 30.0 m. a. Calculate the initial speed of the arrow. (24.2 m/s) b. Calculate the total time the arrow spent in the air. (4.94 s) c. Calculate the time(s) when the arrow is at 15.0 m. (0.73 s and 4.21 s) A person stands on a cliff that is 15.0 m from the ground. He throws a rock at 10.0 m/s straight up, but it falls down off the cliff. a) Calculate the maximum height. (20.1 m) b) Calculate the time up. (1.02 s) c) Calculate the time down (2.03 s) d) Calculate the speed at which it will hit the ground. (-19.8 m/s) A person stands on a building that is 25.0 m from the ground. He throws a rock at 15.0 m/s straight up, but it falls down off the building. a) Calculate the maximum height. (36.5 m) b) Calculate the time up. (1.53 s) c) Calculate the time down (2.73 s) d) Calculate the speed at which it will hit the ground. (-26.7 m/s) 9

10 A person standing on the ground throws a rock at 18.0 m/s. The rock lands on top of a roof that is 10.0 m tall. a) Calculate the maximum height. (16.5 m) b) Calculate the time up. (1.84 s) c) Calculate the velocity at which it will hit the roof. (-11.3 m/s) d) Calculate the time down (1.15 s) Mr. Fredericks standing on the ground throws a student at 21.0 m/s. The student lands on top of a cliff that is 15.0 m tall. a) Calculate the maximum height. (22.5 m) b) Calculate the time up. (2.14 s) c) Calculate the speed at which he will hit the cliff. (-12.1 m/s) d) Calculate the time down (1.24 s) e) Calculate the total distance the student travels. (30.0 m) A student standing on the ground throws Mr. Fredericks at 25.0 m/s. He lands on top of a cliff that is 10.0 m tall. a) Calculate the maximum height. (31.9 m) b) Calculate the time up. (2.55 s) c) Calculate the speed at which he will hit the cliff. (-20.7 m/s) d) Calculate the time down (2.11 s) e) Calculate the total distance he travels. (53.8 m) Graphing Velocity is the slope of the x vs. t graph v = Dx = (x 2 -x 1 ) Dt (t 2 -t 1 ) Acceleration is the slope of the v vs. t graph a = Dv= (v 2 -v 1 ) Dt (t 2 -t 1 ) (Use instantaneous rather than average velocity) (Use instantaneous rather than average velocity) Time (s) Displacement (m) Velocity (m/s) Acceleration (m/s 2 ) Time (s) Displacement (m) Velocity (m/s) Acceleration (m/s 2 ) X 1 22 X X X

11 Time (s) Displacement (m) Velocity (m/s) Acceleration (m/s 2 ) X X (Use instantaneous rather than average velocity) Time (s) Displacement (m) Velocity (m/s) Acceleration (m/s 2 ) X X (Use instantaneous rather than average velocity) 11

12 Acceleration (m/s2) Acceleration vs. Time Time (s) Rules for Graphing Given the following distance-time graph, calculate the velocity of the vehicle. Displacement-time Slope (Derivative) Area (Integrate) Velocity-time Slope (Derivative) Area (Integrate) Acceleration-time Graph the following values time(s) displacement(m) Graphing 12

13 1. Where is the speed the greatest? 2. Where does it turnaround? 3. What is the speed at turnaround? 4. Is it going faster or slower on the way back home? Given the following Velocity versus Time graph: a. Sketch an Acceleration versus Time graph (slope) b. Calculate the displacement (distance) travelled and draw a Displacement versus Time graph. (area) 13

14 Velocity (m/s) Acceleration (m/s2) Displacement (m) Acceleration vs. Time Time (s) Displacement vs. Time Time (s) Given the following a-t graph, draw the velocity-time graph Given the following Velocity versus Time graph: a. Sketch an Acceleration versus Time graph b. Calculate the total distance travelled and sketch a distance-time graph Velocity vs. Time Time(s) 14

15 Displacement (m) Given the following v-t graph, determine the: a. a-t graph b. Displacement after 2 seconds c. Displacement after 6 seconds Velocity(m/s) Time (s) Given the following v-t graph, determine the: a. a-t graph b. Displacement after 100 s 15

16 Mr. Saba is riding his unicorn through the fields. Draw his acceleration-time graph and his distance-time graph. Mr. Saba rides his narwhal at the shore. He starts out away from the beach. 16

17 Given the following graph a) When was he moving away from the beach? b) When was he moving towards beach? c) Where is the turn around point? d) Draw a-t and x-t graphs Kinematic Equations 2) 36 min 8 a) 10.4 m/s b) +3.5 m/s 10) 4.43 h and 881 km/h 14) 5.2 s 16) -6.3 m/s 2 (0.64 g) 20) -2.4 m/s 2 22) a = 4.41 m/s 2, t = 2.61 s 24) 33 m/s 26) a = -390 m/s 2, 40 g s 34) 60.0 m 62) 1.3 m 17

18 34) 60.0 m 36) h = 32 m, t = 5.1 s 38) 16.2 m/s, 13 m 40) v o = 4.85 m/s, t = s 44 a) 12.8 m/s b) 0.735s, 3.35s c) On the way up and on the way down 62) 1.3 m 52.a) constant v is between 0 and 17 s b) t = 28 s c) t = 38 s d) both directions 54 a) 2.5 m/s 2 and 0.6 m/s 2 b) 450 m 56. a) 1700 m b) 500 m Answers to practice quiz 1. a) 4.86 m/s 2 b) 156 m c) 48.6 m/s 2. a) 5.7 m/s b) 0.58 s c) Yes 3. a) 6.25 X m/s 2 b) 7.95 X 10-9 s c) 2.52 X 10 6 m/s 4. a) 1.28 m b) 0.51 s c) 5 m/s d) 11.1 m/s 5. a) 165 m/s 2 b) 330 m c) 165 m/s 6. A) 99.2 m b) 44.1 m/s 7. A) 3.36 m/s 2 b) 1.64 s Answers to Graphing 1 a) 0 b) 5 m/s c) No, distance not changing d) Turns around e) -5 m/s (opposite direction) 2 a) 2 m/s 2 b) 0 m/s 2 c) No, const. v d) -4 m/s 2 e) 400 m f) 600 m 3 a) 55 s b) 55 s onward c) 100 m/s d) 300 m/s e) m/s f) m/s 4 a) Car Y at rest b) 0.5 m/s 2 c) 400 m d) 400 m e) at 40 s 5 a) Scott is moving in the positive direction, Kevin, negative b) 8 hours c) Kevin d) V Kevin = -0.5 cm/hr V Scott = 0.1 cm/hr 37.a = v f 2 - v o 2 2y -9.8 = (0 2 v o2 )/(2 X 2.7) v o = 7.27 m/s a = v f - v o t -9.8 = (0 7.27)/t t up = 0.74 s t total = 2 X 0.74 = 1.48 s 18

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21 Problem 3 You drive for miles at miles/hour. It begins to rain and you slow down to miles/hour. You drive for a total of 3 hours and 20 minutes. a) How far did you travel? b) Calculate your average speed. There s a test TODAY???? 21