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## Kinematics in One Dimension

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**Kinematics in One Dimension**Chapter 2**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**Dynamicsdeals 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**2.1 Displacement**Problem: What is the displacement?**2.1 Displacement**Problem: What is the displacement?***Notice***• 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!***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 thetime required to cover the distance. 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?**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) 15 minutes b) 18 minutes c) 20 minutes d) 22 minutes e) 25 minutes**2.2 Speed and Velocity**Average velocity is the displacement divided by the elapsed time. 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 Jet-Engine 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.**2.2.2. Which one of the following quantities is defined as**the distance traveled divided by the elapsed time for the travel? a) average speed b) average velocity c) average acceleration d) instantaneous velocity e) 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) average speed b) average velocity c) average acceleration d) instantaneous velocity e) instantaneous acceleration**2.3 Acceleration**Accelerationoccurs 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. e) A ball is thrown upward from the surface of the earth, slows to a temporary stop at a height of 4 m, and begins to fall back toward the ground.**2.3 Acceleration**DEFINITION OF AVERAGE ACCELERATIONDistance divided by time2**2.3 Acceleration**Example 3 Acceleration and Increasing Velocity Determine the average acceleration of the plane.**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**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 from24 m/sto 33 m/s. c) A car is traveling southward as its speed decreases from 24m/s to 18 m/s. d) A car is traveling northward as its speed decreases from 24m/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**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 minussign.**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**2.4 Equations of Kinematics for Constant Acceleration**How far does the boat travel?**2.4 Equations of Kinematics for Constant Acceleration**Example 6 Catapulting a Jet Find the displacement of the jet**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?