Chapter 2 Kinematics in One Dimension Kinematics deals with the concepts that are needed to describe motion. Dynamics deals with the effect that forces have on motion. Together, kinematics and dynamics form the branch of physics known as Mechanics. • Scalars are quantities which are fully described by a magnitude alone. (think of “how much”) • Vectors are quantities which are fully described by both a magnitude and a direction. (think of “which way – direction”) • • • • • • • Check your understanding….. 5m 30 m/sec, East 5 mi., North 20 degrees Celsius 256 bytes 4000 Calories Lesson 1: Describing Motion with Words Introduction to the Language of Kinematics • Distance is a scalar quantity which refers to "how much ground an object has covered" during its motion. • Displacement is a vector quantity which refers to "how far out of place an object is"; it is the object's change in position. 2.1 Displacement x final position xo initialposition x x xo displacement Lesson 1: Describing Motion with Words Introduction to the Language of Kinematics • Distance is a scalar quantity which refers to "how much ground an object has covered" during its motion. • Displacement is a vector quantity which refers to "how far out of place an object is"; it is the object's change in position. 2.1 Displacement xo 2.0 m x 5.0 m x 7.0 m x x xo 7.0 m 2.0 m 5.0 m 2.1 Displacement x 2.0 m x 5.0 m xo 7.0 m x x xo 2.0 m 7.0 m 5.0 m 2.1 Displacement xo 2.0 m x 5.0 m x 7.0 m x x xo 5.0 m 2.0 m 7.0 m Lesson 1: Describing Motion with Words Speed and Velocity • Speed is a scalar quantity which refers to "how fast an object is moving." A fast-moving object has a high speed while a slow-moving object has a low speed. An object with no movement at all has a zero speed. • Velocity is a vector quantity which refers to "the rate at which an object changes its position." 2.2 Speed and Velocity Average speed is the distance traveled divided by the time required to cover the distance. Distance Average speed Elapsed time SI units for speed: meters per second (m/s) 2.2 Speed and Velocity Example 1 Distance Run by a Jogger How far does a jogger run in 1.5 hours if his average speed is 2.22 m/s? Distance Average speed Elapsed time 2.2 Speed and Velocity Average velocity is the displacement divided by the elapsed time. Displacement Average velocity Elapsed time x x o x v t to t 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 Speed and Velocity x 1609 m v 339 .5 m s t 4.740 s x 1609 m v 342 .7 m s t 4.695 s Constant Speed problems A horse canters away from its trainer in a straight line moving 100. m away in 16.0 s. It then turns abruptly and gallops halfway back in 4.6. Calculate the average speed and average velocity. A bike travels at a constant speed of 4.0 m/s for 5 s. How far does it go? The round trip distance between Earth and the moon is 350,000 km, if the speed of a laser is 3.0 x 108 m/s how much time does it take the laser to travel from Earth to the moon? 2.3 Acceleration The notion of acceleration emerges when a change in velocity is combined with the time during which the change occurs. 2.3 Acceleration DEFINITION OF AVERAGE ACCELERATION v v o v a t to t 2.3 Acceleration Example 3 Acceleration and Increasing Velocity Determine the average acceleration of the plane. vo 0 m s v 260km h to 0 s t 29 s 2.3 Acceleration 2.3 Acceleration Example 3 Acceleration and Decreasing Velocity v v o 13m s 28m s a 5.0 m s 2 t to 12 s 9 s 2.4 Equations of Kinematics for Constant Acceleration Equations of Kinematics for Constant Acceleration v vo at x 1 2 vo v t v v 2ax 2 2 o x vot at 1 2 2 2.4 Equations of Kinematics for Constant Acceleration Five kinematic variables: 1. displacement, x 2. acceleration (constant), a 3. final velocity (at time t), v 4. initial velocity, vo 5. elapsed time, t 2.4 Equations of Kinematics for Constant Acceleration x vot at 1 2 2 6.0 m s 8.0 s 110 m 1 2 2.0 m s 8.0 s 2 2 2.4 Equations of Kinematics for Constant Acceleration Example 6 Catapulting a Jet Find its displacement. vo 0 m s x ?? a 31m s2 v 62 m s 2.4 Equations of Kinematics for Constant Acceleration v v 62 m s 0 m s x 62 m 2 2a 2 31m s 2 2 o 2 2 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 x a v vo +215000 m -10.0 m/s2 ? +3250 m/s v v 2ax 2 2 o t v v 2ax 2 o v 3250m s 2 10.0 m s 2 215000m 2 2500m s 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 9.80m 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 vot at 1 2 vo t 0 m/s 3.00 s 2 0 m s 3.00 s 44.1 m v 1 2 9.80m s 3.00 s 2 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 y 2a 2 v v 2ay 2 t 2 o 2 o v v 0 m s 5.00 m s y 1.28 m 2 2a 2 9.80 m s 2 2 o 2 2 2.6 Freely Falling Bodies Conceptual Example 14 Acceleration Versus Velocity There are three parts to the motion of the coin. On the way up, the coin has a vector velocity that is directed upward and has decreasing magnitude. At the top of its path, the coin momentarily has zero velocity. On the way down, the coin has downward-pointing velocity with an increasing magnitude. In the absence of air resistance, does the acceleration of the coin, like the velocity, change from one part to another? 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? Position-Time Graphs • We can use a postion-time graph to illustrate the motion of an object. • Postion is on the yaxis • Time is on the x-axis Plotting a Distance-Time Graph • Axis – Distance (position) on y-axis (vertical) – Time on x-axis (horizontal) • Slope is the velocity – Steeper slope = faster – No slope (horizontal line) = staying still Where and When • We can use a position time graph to tell us where an object is at any moment in time. • Where was the car at 4 s? • 30 m • How long did it take the car to travel 20 m? • 3.2 s Interpret this graph… Describing in Words Describing in Words • Describe the motion of the object. • When is the object moving in the positive direction? • Negative direction. • When is the object stopped? • When is the object moving the fastest? • The slowest? Accelerated Motion • In a position/displacement time graph a straight line denotes constant velocity. • In a position/displacement time graph a curved line denotes changing velocity (acceleration). • The instantaneous velocity is a line tangent to the curve. Accelerated Motion • In a velocity time graph a line with no slope means constant velocity and no acceleration. • In a velocity time graph a sloping line means a changing velocity and the object is accelerating. Velocity • Velocity changes when an object… – Speeds Up – Slows Down – Change direction Velocity-Time Graphs • Velocity is placed on the vertical or y-axis. • Time is place on the horizontal or x-axis. • We can interpret the motion of an object using a velocity-time graph. Constant Velocity • Objects with a constant velocity have no acceleration • This is graphed as a flat line on a velocity time graph. Changing Velocity • Objects with a changing velocity are undergoing acceleration. • Acceleration is represented on a velocity time graph as a sloped line. Positive and Negative Velocity • The first set of graphs show an object traveling in a positive direction. • The second set of graphs show an object traveling in a negative direction. Speeding Up and Slowing Down •The graphs on the left represent an object speeding up. •The graphs on the right represent an object that is slowing down. Two Stage Rocket • Between which time does the rocket have the greatest acceleration? • At which point does the velocity of the rocket change. Displacement from a Velocity-Time Graph • The shaded region under a velocity time graph represents the displacement of the object. • The method used to find the area under a line on a velocity-time graph depends on whether the section bounded by the line and the axes is a rectangle, a triangle 2.7 Graphical Analysis of Velocity and Acceleration x 8 m Slope 4 m s t 2s 2.7 Graphical Analysis of Velocity and Acceleration 2.7 Graphical Analysis of Velocity and Acceleration v 12 m s Slope 6 m s 2 t 2s

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