Chapter 2 Kinematics in One Dimension Objectives • We will compare and contrast distance to displacement, and speed to velocity • We will be able to solve problems using distance, displacement, speed and velocity Kinematics deals with the concepts that are needed to describe motion. These concepts are: Displacement Velocity Acceleration Time Dynamics deals with the effect that forces have on motion. Together, kinematics and dynamics form the branch of physics known as Mechanics. Right now we are focused on Kinematics, how things move. Not what is pushing them to move (dynamics) 2.1 Displacement Displacement is the shortest distance from the initial to the final position x o initial position x final position x x x o displaceme nt 2.1 Displacement Problem: What is the displacement? x o 2 .0 m x 5 .0 m x 7 .0 m x x x o 7.0 m 2.0 m 5 . 0 m 2.1 Displacement Problem: What is the displacement? x 2 .0 m x 5 .0 m x o 7 .0 m r r r x x x o 2.0 m 7.0 m 5 .0 m *Notice* r r r x x x o 2.0 m 7.0 m 5 .0 m • In the last problem the answer is -5.0m. What does that mean? • It means that the object traveled in the negative direction 5.0m. IT DOES NOT MEAN THE OBJECT WAS WALKING BACKWARDS OR GOING BACK IN TIME! 2.1 Displacement x o 2 .0 m x 5 .0 m x 7 .0 m x x x o 5.0 m 2 .0 m 7 . 0 m *Distance is not Displacement* • Distance refers to the total amount of land covered (For Example: you walk around a track and you have covered 400 m.) • Displacement is the final point – the initial point (For Example: you walk around a track and your displacement is 0 m.) 2.1.1. The branch of physics that deals with motion is called mechanics. Kinematics is the portion of mechanics that describes motion without any reference to which of the following concepts? a) forces b) accelerations c) velocities d) displacements e) time 2.1.2. A particle travels along a curved path between two points A and B as shown. Complete the following statement: The displacement of the particle does not depend on a) the location of A. b) the location of B. c) the direction of A from B. d) the distance traveled from A to B. e) the shortest distance between A and B. 2.1.3. For which one of the following situations will the path length equal the magnitude of the displacement? a) An Olympic athlete is running around an oval track. b) A roller coaster car travels up and down two hills. c) A truck travels 4 miles west; and then, it stops and travels 2 miles west. d) A ball rises and falls after being thrown straight up from the earth's surface. e) A ball on the end of a string is moving in a vertical circle. 2.2 Speed and Velocity Average speed is the distance traveled divided by the time required to cover the distance. Average speed Distance Elapsed time SI units for speed: meters per second (m/s) Try to let go of miles per hour! 2.2 Speed and Velocity Example 1 Distance Run by a Jogger How far does a jogger run in 1.5 hours (5400 s) if his average speed is 2.22 m/s? Average speed Distance Elapsed Distance Average speed time Elapsed 2 . 22 m s 5400 s 12000 m time 2.2.1. A turtle and a rabbit are to have a race. The turtle’s average speed is 0.9 m/s. The rabbit’s average speed is 9 m/s. The distance from the starting line to the finish line is 1500 m. The rabbit decides to let the turtle run before he starts running to give the turtle a head start. What, approximately, is the maximum time the rabbit can wait before starting to run and still win the race? a) b) c) d) e) 15 minutes 18 minutes 20 minutes 22 minutes 25 minutes 2.2 Speed and Velocity Average velocity is the displacement divided by the elapsed time. Average velocity Displaceme Elapsed v x xo t to nt time x t Remember – Its ALWAYS FINAL minus INITIAL. Even if the number turns out to be NEGATIVE 2.2 Speed and Velocity Example 2 The World’s Fastest JetEngine Car Andy Green in the car ThrustSSC set a world record of 341.1 m/s in 1997. To establish such a record, the driver makes two runs through the course, one in each direction, to nullify wind effects. From the data, determine the average velocity for each run. 2.2 Speed and Velocity v v x t x t 1609 m 339 . 5 m s 4.740 s 1609 m 4.695 s 342 . 7 m s 2.2.2. Which one of the following quantities is defined as the distance traveled divided by the elapsed time for the travel? a) b) c) d) e) average speed average velocity average acceleration instantaneous velocity instantaneous acceleration 2.2.3. Which one of the following quantities is defined as an object’s displacement divided by the elapsed time for the displacement? a) b) c) d) e) average speed average velocity average acceleration instantaneous velocity instantaneous acceleration 2.3 Acceleration Acceleration occurs when there is a change in velocity during a specific time period 2.3.1. Which one of the following situations does the object have no acceleration? a) A ball at the end of a string is whirled in a horizontal circle at a constant speed. b) Seeing a red traffic light ahead, the driver of a minivan steps on the brake. As a result, the minivan slows from 15 m/s to stop before reaching the light. c) A boulder starts from rest and rolls down a mountain. d) An elevator in a tall skyscraper moves upward at a constant speed of 3 m/s. 2.3 Acceleration DEFINITION OF AVERAGE ACCELERATION a v vo t to v t Distance divided by time2 2.3 Acceleration Example 3 Acceleration and Increasing Velocity Determine the average acceleration of the plane. vo 0 m s a v vo t to v 260 km h to 0 s 260 km h 0 km h 29 s 0 s t 29 s 9 .0 km h s 2.3 Acceleration 2.3.4. A sports car starts from rest. After 10.0 s, the speed of the car is 25.0 m/s. What is the magnitude of the car’s acceleration? a) 2.50 m/s2 b) 5.00 m/s2 c) 10.0 m/s2 d) 25.0 m/s2 e) 250 m/s2 2.3 Acceleration Example 3 – Acceleration and Decreasing Velocity Solve for acceleration a v vo t to 13 m s 28 m s 12 s 9 s 5 .0 m s 2 2.3 Acceleration Common Usage • If an object is slowing down it is still “accelerating” because the velocity is changing. • However, most people refer to that as “decelerating” 2.3.2. In which one of the following situations does the car have an acceleration that is directed due north? a) A car travels northward with a constant speed of 24 m/s. b) A car is traveling southward as its speed increases from 24 m/s to 33 m/s. c) A car is traveling southward as its speed decreases from 24 m/s to 18 m/s. d) A car is traveling northward as its speed decreases from 24 m/s to 18 m/s. e) A car travels southward with a constant speed of 24 m/s. 2.3.3. A postal truck driver driving due east gently steps on her brake as she approaches an intersection to reduce the speed of the truck. What is the direction of the truck’s acceleration, if any? a) There is no acceleration in this situation. b) due north c) due east d) due south e) due west Question • How many “Accelerators” does a car have? • • • • 3 Gas pedal Brake Steering Wheel – A change in direction is a change in velocity 2.3.4. The drawing shows the position of a rolling ball at one second intervals. Which one of the following phrases best describes the motion of this ball? a) constant position b) constant velocity c) increasing velocity d) constant acceleration e) decreasing velocity 2.3.5. A police cruiser is parked by the side of the road when a speeding car passes. The cruiser follows the speeding car. Consider the following diagrams where the dots represent the cruiser’s position at 0.5-s intervals. Which diagram(s) are possible representations of the cruiser’s motion? a) A only b) B, D, or E only c) C only d) E only e) A or C only Some Minutia • So far we have analyzed average velocity, speed and acceleration. • Instantaneous speed, velocity or acceleration is the speed, velocity or acceleration of an object at a specific time. For example – Speedometer gives us instantaneous speed. Easy Rule of Thumb or Hands – HA! • Velocity – Right Hand, Acceleration – Left • Arms together – object speeding up • Arms separate – object slowing down 2.4.1. Complete the following statement: For an object moving at constant acceleration, the distance traveled a) increases for each second that the object moves. b) is the same regardless of the time that the object moves. c) is the same for each second that the object moves. d) cannot be determined, even if the elapsed time is known. e) decreases for each second that the object moves. 2.4.2. Complete the following statement: For an object moving with a negative velocity and a positive acceleration, the distance traveled a) increases for each second that the object moves. b) is the same regardless of the time that the object moves. c) is the same for each second that the object moves. d) cannot be determined, even if the elapsed time is known. e) decreases for each second that the object moves. 2.4.3. At one particular moment, a subway train is moving with a positive velocity and negative acceleration. Which of the following phrases best describes the motion of this train? Assume the front of the train is pointing in the positive x direction. a) The train is moving forward as it slows down. b) The train is moving in reverse as it slows down. c) The train is moving faster as it moves forward. d) The train is moving faster as it moves in reverse. e) There is no way to determine whether the train is moving forward or in reverse. 2.4 Equations of Kinematics for Constant Acceleration v x xo t to a v vo t to It is customary to dispense with the use of boldface Symbols overdrawn with arrows for the displacement, velocity, and acceleration vectors. We will, however, continue to convey the directions with a plus or minus sign. v x xo t to a v vo t to Objectives We will be able to differentiate and use the 4 equations of kinematics to solve kinematic problems 2.4 Equations of Kinematics for Constant Acceleration Equations of Kinematics for Constant Acceleration v v o at x 1 2 v o v t v v 2 ax 2 2 o x x o v ot 1 2 at 2 2.4 Equations of Kinematics for Constant Acceleration How far does the boat travel? x x o v ot 1 2 at 2 0 m + 6 .0 m s 8 .0 s 110 m 1 2 2 .0 m s 2 8 .0 s 2 2.4 Equations of Kinematics for Constant Acceleration Example 6 Catapulting a Jet Find the displacement of the jet vo 0 m s x ?? a 31 m s v 62 m s 2 2.4 Equations of Kinematics for Constant Acceleration v v 2 x 2a 2 o 62 m s 2 0 m s 2 31 m s 2 2 62 m 2.5 Applications of the Equations of Kinematics Reasoning Strategy 1. Make a drawing. 2. Decide which directions are to be called positive (+) and negative (-). 3. Write down the values that are given for any of the five kinematic variables. 4. Verify that the information contains values for at least three of the five kinematic variables. Select the appropriate equation. 5. When the motion is divided into segments, remember that the final velocity of one segment is the initial velocity for the next. 6. Keep in mind that there may be two possible answers to a kinematics problem. 2.5 Applications of the Equations of Kinematics Example 8 An Accelerating Spacecraft A spacecraft is traveling with a velocity of +3250 m/s. Suddenly the retrorockets are fired, and the spacecraft begins to slow down with an acceleration whose magnitude is 10.0 m/s2. What is the velocity of the spacecraft when the displacement of the craft is +215 km, relative to the point where the retrorockets began firing? x a v vo +215000 m -10.0 m/s2 ? +3250 m/s t 2.5 Applications of the Equations of Kinematics 2.5 Applications of the Equations of Kinematics x a v vo +215000 m -10.0 m/s2 ? +3250 m/s v v 2 ax 2 v 2 o 3250 2500 m s v m s 2 10 . 0 m s 2 t v 2 ax 2 o 2 215000 m 2.5.1. Starting from rest, two objects accelerate with the same constant acceleration. Object A accelerates for three times as much time as object B, however. Which one of the following statements is true concerning these objects at the end of their respective periods of acceleration? a) Object A will travel three times as far as object B. b) Object A will travel nine times as far as object B. c) Object A will travel eight times as far as object B. d) Object A will be moving 1.5 times faster than object B. e) Object A will be moving nine times faster than object B. 2.6 Freely Falling Bodies In the absence of air resistance, it is found that ALL bodies at the same location above the Earth fall vertically with the same acceleration. This idealized motion is called free-fall and the acceleration of a freely falling body is called the acceleration due to gravity. g 9 . 80 m s 2 or 32 . 2 ft s 2 2.6 Freely Falling Bodies g 9 . 80 m s 2 2.6 Freely Falling Bodies Example 10 A Falling Stone A stone is dropped from the top of a tall building. After 3.00s of free fall, what is the displacement y of the stone? 2.6 Freely Falling Bodies y a ? -9.80 m/s2 v vo t 0 m/s 3.00 s 2.6 Freely Falling Bodies y a ? -9.80 m/s2 y y o v ot 1 2 at vo t 0 m/s 3.00 s 2 0 m 0 m s 3 .00 s 44 .1 m v 1 2 9 .80 m s 2 3 .00 s 2 2.6 Freely Falling Bodies Example 12 How High Does it Go? The referee tosses the coin up with an initial speed of 5.00m/s. In the absence if air resistance, how high does the coin go above its point of release? 2.6 Freely Falling Bodies y a v vo ? -9.80 m/s2 0 m/s +5.00 m/s t 2.6 Freely Falling Bodies y a v vo ? -9.80 m/s2 0 m/s +5.00 m/s v v 2 v v 2 ay 2 2 o v v 2 y 2a y t 2 o 2a 2 o 0 m s 2 5 . 00 m s 2 9 . 80 m s 2 2 1 . 28 m 2.6 Freely Falling Bodies Conceptual Example 14 Acceleration Versus Velocity The following picture shows the path of the coin from the previous problem with velocity vectors. On this picture. Draw acceleration vectors. 2.6 Freely Falling Bodies Conceptual Example 15 Taking Advantage of Symmetry Does the pellet in part b strike the ground beneath the cliff with a smaller, greater, or the same speed as the pellet in part a? 2.6 Freely Falling Bodies Conceptual Example 15 Taking Advantage of Symmetry Same speed as part A. In the absence of air resistance, an object fired in the air at a certain velocity, vo, will have the same speed, vo, when it returns to the same height at which it as fired. Another side note • In free fall problems if an object is moving up, at the top of its path the velocity is always 0 m/s. However the acceleration is still 9.8m/s2. 2.6.1. A rock is released from rest from a hot air balloon that is at rest with respect to the ground a few meters below. If we ignore air resistance as the rock falls, which one of the following statements is true? a) The rock will take longer than one second to reach the ground. b) The instantaneous speed of the rock just before it reaches the ground will be 9.8 m/s. c) The rock is considered a freely falling body after it is released. d) As the rock falls, its acceleration is 9.8 m/s2, directed upward. e) After the ball is released it falls at a constant speed of 9.8 m/s. 2.6.2. Ping-pong ball A is filled with sand. Ping-pong ball B is identical to A, except that it is empty inside. Ball A is somewhat heavier than ball B because of the sand inside. Both balls are simultaneously dropped from rest from the top of a building. Which of these two balls has the greater acceleration due to gravity, if any, as they fall? a) b) c) d) ball A ball B Both ball A and ball B have zero acceleration. Both ball A and ball B have the same acceleration. 2.6.3. A ball is thrown vertically upward from the surface of the earth. The ball rises to some maximum height and falls back toward the surface of the earth. Which one of the following statements concerning this situation is true if air resistance is neglected? a) As the ball rises, its acceleration vector points upward. b) The ball is a freely falling body for the duration of its flight. c) The acceleration of the ball is zero when the ball is at its highest point. d) The speed of the ball is negative while the ball falls back toward the earth. e) The velocity and acceleration of the ball always point in the same direction. Objectives We will be able to describe the motion of an object when given a position, velocity or acceleration graph When given one of the graphs, we will be able to draw the other 2 graphs. 2.7 Graphical Analysis of Velocity and Acceleration – Most important part of chapter! • What is the definition of slope? • Rise / run or Δy/Δx 2.7 Graphical Analysis of Velocity and Acceleration – Most important part of chapter! • What is the slope of the graph below? 2.7 Graphical Analysis of Velocity and Acceleration Slope x t 8m 2s 4 m s Notice… • On the graph, the y-axis is position and the x-axis is time. • Slope is Δy/Δx or position over time • The slope of a position-time graph IS THE VELOCITY!! 2.7 Graphical Analysis of Velocity and Acceleration What is the velocity for each part of the graph? More Mystery Walk 1 Now Sketch a velocity vs time graph for Mystery Walk 1. Velocity vs Time Graph Walk 1 Remember! Velocity is Speed in a particular direction. Our walker reversed direction to return to the starting point. Mystery Walk 2 Based on the graph, describe what the walker did. Mystery Walk 2 He waited for 4 seconds before starting to walk slowly at constant velocity Mystery Walk 2 Now, sketch a velocity time graph of Mystery Walk 2. Mystery Walk 2 Velocity vs Time Graph Mystery Walk 3 Based on the graph describe what the walker did during her walk. Mystery Walk 3 She walked backward very slowly. After 5 seconds, she ran forward for 5 more seconds. Mystery Walk 3 Sketch a Velocity vs Time graph of Mystery Walk 3. Velocity vs Time for Mystery Walk 3 Which v vs t Graph Corresponds to the Given d vs t Graph? ? Which v vs t Graph Corresponds to the Given d vs t Graph? ? 2.7.1. A dog is initially walking due east. He stops, noticing a cat behind him. He runs due west and stops when the cat disappears into some bushes. He starts walking due east again. Then, a motorcycle passes him and he runs due east after it. The dog gets tired and stops running. Which of the following graphs correctly represent the position versus time of the dog? 2.7 Graphical Analysis of Velocity and Acceleration •Another Important point •The area UNDER a velocity-time graph is equal to the DISPLACEMENT of the object 2.7 Graphical Analysis of Velocity and Acceleration •Notice is the graph below the velocity is not constant because the slope is not constant. •In the graph below the velocity is changing, therefore the object is ACCELERATING! 2.7 Graphical Analysis of Velocity and Acceleration 2.7 Graphical Analysis of Velocity and Acceleration •Another Important point •The area UNDER a acceleration-time graph is equal to the velocity of the object Which v vs t Graph Corresponds to the Given d vs t Graph? ? Which v vs t Graph Corresponds to the Given d vs t Graph? ? 2.7.3. Consider the graph the position versus time graph shown. Which curve on the graph best represents a constantly accelerating car? a) A b) B c) C d) D e) None of the curves represent a constantly accelerating car. 2.7.3. The graph shows the velocity of an object versus the elapsed time. During which interval on the graph does the object’s acceleration decrease with time? a) A b) B c) C d) D e) E 2.7.4. Complete the following statement: the instantaneous acceleration of an object can be determined by determining the slope of a) the object’s velocity versus elapsed time graph. b) the object’s displacement versus elapsed time graph. c) the object’s distance versus elapsed time graph. d) the object’s acceleration versus elapsed time graph. 2.7 Graphical Analysis of Velocity and Acceleration Slope v t 12 m s 2s 6 m s 2 Notice… • On the graph, the y-axis is velocity and the x-axis is time. • Slope is Δy/Δx or velocity over time • The slope of a velocity-time graph IS THE ACCELERATION!! x All 3 Graphs t v t a t Graphical Comparison Given the displacementtime graph (a) The velocity-time graph is found by measuring the slope of the positiontime graph at every instant. The acceleration-time graph is found by measuring the slope of the velocity-time graph at every instant. Section 2.4 Graph Practice- On Board Try making all three graphs for the following scenario: 1. Schmedrick starts out north of home. At time zero he’s driving a cement mixer south very fast at a constant speed. 2. He accidentally runs over an innocent moose crossing the road, so he slows to a stop to check on the poor moose. 3. He pauses for a while until he determines the moose is squashed flat and deader than a doornail. 4. Fleeing the scene of the crime, Schmedrick takes off again in the same direction, speeding up quickly. 5. When his conscience gets the better of him, he slows, turns around, and returns to the crash site.