© Houghton Mifflin Harcourt Publishing Company Preview Objectives One Dimensional Motion Displacement Average Velocity Velocity and Speed Interpreting Velocity Graphically Chapter 2 Section 1 Displacement and Velocity

© Houghton Mifflin Harcourt Publishing Company Section 1 Displacement and Velocity Chapter 2 Objectives Describe motion in terms of frame of reference, displacement, time, and velocity. Calculate the displacement of an object traveling at a known velocity for a specific time interval. Construct and interpret graphs of position versus time.

© Houghton Mifflin Harcourt Publishing Company Section 1 Displacement and Velocity Chapter 2 One Dimensional Motion To simplify the concept of motion, we will first consider motion that takes place in one direction. One example is the motion of a commuter train on a straight track. (can only move forward or backward; not left or right) To measure motion, you must choose a frame of reference. A frame of reference is a system for specifying the precise location of objects in space and time. (remains the same for a problem; has starting point from which motion is measured)

© Houghton Mifflin Harcourt Publishing Company Click below to watch the Visual Concept. Visual Concept Chapter 2 Section 1 Displacement and Velocity Frame of Reference Choose the frame of reference that makes explaining things easiest for you!

© Houghton Mifflin Harcourt Publishing Company Section 1 Displacement and Velocity Chapter 2 Displacement  x = x f – x i displacement = final position – initial position Displacement is a change in position. Length of a STRAIGHT line path between the initial and final position The SI unit of displacement is the meter, m.

© Houghton Mifflin Harcourt Publishing Company Displacement Displacement is not always equal to zero; if a object starts and stops at the same position then it has a displacement of zero Displacement can be positive or negative –Movement to the left and down will be deemed negative –Movement to the right and up will be deemed positive

© Houghton Mifflin Harcourt Publishing Company Click below to watch the Visual Concept. Visual Concept Chapter 2 Section 1 Displacement and Velocity Displacement

© Houghton Mifflin Harcourt Publishing Company Chapter 2 Positive and Negative Displacements Section 1 Displacement and Velocity

© Houghton Mifflin Harcourt Publishing Company Section 1 Displacement and Velocity Chapter 2 Average Velocity Average velocity is the total displacement divided by the time interval during which the displacement occurred. In SI, the unit of velocity is meters per second, abbreviated as m/s.

© Houghton Mifflin Harcourt Publishing Company Average Velocity Can be positive or negative Equal to the constant velocity needed to cover a given displacement in a given time interval –May have been times during trip when traveling faster or slower Average velocity is NOT the average of the starting and ending velocities!

© Houghton Mifflin Harcourt Publishing Company Click below to watch the Visual Concept. Visual Concept Chapter 2 Section 1 Displacement and Velocity Average Velocity

© Houghton Mifflin Harcourt Publishing Company Example Problem 1.A doctor travels to the east from city A to city B (75 km) in 1.0 h. What is the doctor’s average velocity? 2.Heather and Matthew walk with an average velocity of 0.98 m/s eastward. If it takes them 34 minutes to walk to the store, what is their displacement? 3.If Joe rides his bicycle in a straight line for 15 minutes with an average velocity of 12.5 km/h south, how far has he ridden? 4.It takes you 9.5 minutes to walk with an average velocity of 1.2 m/s to the north from the bust stop to the museum entrance. What is your displacement?

© Houghton Mifflin Harcourt Publishing Company Example Problems Continued 5.Simpson drives his car with an average velocity of 48.0 km/h to the east. How long will it take him to drive 144 km on a straight highway? a.How much time would Simpson save by increasing his average velocity to 56.0 km/h to the east? 6.A bus travels 280 km south along a straight path with an average velocity of 88 km/h to the south. The bus stops for 24 minutes. Then, it travels 210 km south with an average velocity of 75 km/h to the south. a.How long does the total trip last? b.What is the average velocity for the total trip?

© Houghton Mifflin Harcourt Publishing Company Section 1 Displacement and Velocity Chapter 2 Velocity and Speed Velocity describes motion with both a direction and a numerical value (a magnitude). Speed has no direction, only magnitude. Average speed is equal to the total distance traveled divided by the time interval.

© Houghton Mifflin Harcourt Publishing Company Section 1 Displacement and Velocity Chapter 2 Interpreting Velocity Graphically –Object 1: positive slope = positive velocity –Object 2: zero slope= zero velocity –Object 3: negative slope = negative velocity For any position-time graph, we can determine the average velocity by drawing a straight line between any two points on the graph. If the velocity is constant, the graph of position versus time is a straight line. The slope indicates the velocity.

© Houghton Mifflin Harcourt Publishing Company Section 1 Displacement and Velocity Chapter 2 Interpreting Velocity Graphically, continued The instantaneous velocity at a given time can be determined by measuring the slope of the line that is tangent to that point on the position-versus-time graph. The instantaneous velocity is the velocity of an object at some instant or at a specific point in the object’s path.

© Houghton Mifflin Harcourt Publishing Company Click below to watch the Visual Concept. Visual Concept Chapter 2 Section 1 Displacement and Velocity Sign Conventions for Velocity

© Houghton Mifflin Harcourt Publishing Company Homework Formative Assessment pg. 43 #1-6

© Houghton Mifflin Harcourt Publishing Company Preview Objectives Changes in Velocity Motion with Constant Acceleration Sample Problem Chapter 2 Section 2 Acceleration

© Houghton Mifflin Harcourt Publishing Company Chapter 2 Objectives Describe motion in terms of changing velocity. Compare graphical representations of accelerated and nonaccelerated motions. Apply kinematic equations to calculate distance, time, or velocity under conditions of constant acceleration. Section 2 Acceleration

© Houghton Mifflin Harcourt Publishing Company Chapter 2 Changes in Velocity Acceleration is the rate at which velocity changes over time. An object accelerates if its speed, direction, or both change. Acceleration has direction and magnitude. Thus, acceleration is a vector quantity. Section 2 Acceleration

© Houghton Mifflin Harcourt Publishing Company Click below to watch the Visual Concept. Visual Concept Chapter 2 Section 2 Acceleration Acceleration SI units of m/s 2

© Houghton Mifflin Harcourt Publishing Company Acceleration Problem Clues “slows down” – acceleration is opposition direction of motion “speeds up” – acceleration is in direction of motion “starts from rest” – initial velocity is 0 m/s “complete stop and comes to rest” – final velocity is 0 m/s “uniform acceleration” – acceleration is NOT changing “uniform velocity” – velocity is NOT changing (i.e. 0 acceleration)

© Houghton Mifflin Harcourt Publishing Company Example Problems 1.Find the acceleration of an amusement park ride that falls from rest to a speed of 28 m/s in 3.0 s. 2.As the shuttle bus comes to a sudden stop to avoid hitting a dog, it accelerates uniformly at -4.1 m/s 2 as it slows from 9.0 m/s to 0.0 m/s. Find the time interval of acceleration for the bus. 3.A car traveling at 7.0 m/s accelerates uniformly at 2.5 m/s 2 to reach a speed of 12 m/s. How long does it take for this acceleration to occur?

© Houghton Mifflin Harcourt Publishing Company Example Problems Continued 4.With an average acceleration of -1.2 m/s 2, how long will it take a cyclist to bring a bicycle with an initial speed of 6.5 m/s to a complete stop? 5.Turner’s treadmill runs with a velocity of -1.2 m/s and speeds up at regular intervals during a half hour workout. After 25 minutes, the treadmill has a velocity of -6.5 m/s. What is the average acceleration of the treadmill during this period? 6.Suppose a treadmill has an average acceleration of 4.7 x m/s 2. a.How much does it change its speed after 5 minutes b.If the treadmill’s initial speed is 1.7 m/s, what will its final speed be?

© Houghton Mifflin Harcourt Publishing Company Chapter 2 Changes in Velocity, continued Consider a train moving to the right, so that the displacement and the velocity are positive. The slope of the velocity-time graph is the average acceleration. Section 2 Acceleration –When the velocity in the positive direction is increasing, the acceleration is positive, as at A. –When the velocity is constant, there is no acceleration, as at B. –When the velocity in the positive direction is decreasing, the acceleration is negative, as at C.

© Houghton Mifflin Harcourt Publishing Company Click below to watch the Visual Concept. Visual Concept Chapter 2 Section 2 Acceleration Graphical Representations of Acceleration

© Houghton Mifflin Harcourt Publishing Company Chapter 2 Velocity and Acceleration Section 2 Acceleration Use this table to check your answers for problems involving acceleration

© Houghton Mifflin Harcourt Publishing Company Chapter 2 Motion with Constant Acceleration When velocity changes by the same amount during each time interval, acceleration is constant. The relationships between displacement, time, velocity, and constant acceleration are expressed by the equations shown on the next slide. These equations apply to any object moving with constant acceleration. These equations use the following symbols:  x = displacement v i = initial velocity v f = final velocity  t = time interval Section 2 Acceleration

© Houghton Mifflin Harcourt Publishing Company Chapter 2 Equations for Constantly Accelerated Straight-Line Motion Section 2 Acceleration

© Houghton Mifflin Harcourt Publishing Company Sample Problem Final Velocity After Any Displacement A person pushing a stroller starts from rest, uniformly accelerating at a rate of m/s 2. What is the velocity of the stroller after it has traveled 4.75 m? Chapter 2 Section 2 Acceleration

© Houghton Mifflin Harcourt Publishing Company Sample Problem, continued 1. Define Given: v i = 0 m/s a = m/s 2  x = 4.75 m Unknown: v f = ? Diagram: Choose a coordinate system. The most convenient one has an origin at the initial location of the stroller, as shown above. The positive direction is to the right. Chapter 2 Section 2 Acceleration

© Houghton Mifflin Harcourt Publishing Company Chapter 2 Sample Problem, continued 2. Plan Choose an equation or situation: Because the initial velocity, acceleration, and displacement are known, the final velocity can be found using the following equation: Rearrange the equation to isolate the unknown: Take the square root of both sides to isolate v f. Section 2 Acceleration

© Houghton Mifflin Harcourt Publishing Company Chapter 2 Sample Problem, continued Tip: Think about the physical situation to determine whether to keep the positive or negative answer from the square root. In this case, the stroller starts from rest and ends with a speed of 2.18 m/s. An object that is speeding up and has a positive acceleration must have a positive velocity. So, the final velocity must be positive. 3. Calculate Substitute the values into the equation and solve: 4. Evaluate The stroller’s velocity after accelerating for 4.75 m is 2.18 m/s to the right. Section 2 Acceleration

© Houghton Mifflin Harcourt Publishing Company Example Problems 1.A car accelerates uniformly from rest to a speed of 6.6 m/s in 6.5 s. Find the distance the car travels during this time. 2.When Maggie applies the brakes of her car, the car slows uniformly from 15 m/s to 0.0 m/s in 2.50 s. How many meters before a stop sign must she apply her brakes in order to stop at the sign? 3.A driver in a car traveling at a speed of 21.8 m/s sees a cat 101 m away on the road. How long will it take for the car to accelerate uniformly to a stop in exactly 99 m?

© Houghton Mifflin Harcourt Publishing Company Example Problems Continued 4.A car enters the freeway with a speed of 6.4 m/s and accelerates uniformly for 3.2 km in 3.5 min. How fast (in m/s) is the car moving after this time? 5.A car with an initial speed of 6.5 m/s accelerates at a uniform rate of 0.92 m/s 2 for 3.6 s. Find the final speed and the displacement of the car during this time. 6.An automobile with an initial speed of 4.30 m/s accelerates uniformly at the rate of 3.00 m/s 2. Find the final speed and the displacement after 5.00 s.

© Houghton Mifflin Harcourt Publishing Company Example Problems Continued 7.A car starts from rest and travels for 5.0 s with a constant acceleration of -1.5 m/s 2. What is the final velocity of the car? How far does the car travel in this time interval? 8.A driver of a car traveling at 15.0 m/s applies the brakes, causing a uniform acceleration of -2.0 m/s 2. How long does it take the car to accelerate to a final speed of 10 m/s? How far has the car moved during the braking period?

© Houghton Mifflin Harcourt Publishing Company Example Problems Continued 9.A car traveling initially at 7.0 m/s accelerates uniformly at a rate of 0.80 m/s 2 for a distance of 245 m. a.What is the velocity at the end of the acceleration? b.What is its velocity after it accelerates for 125 m? c.What is its velocity after it accelerates for 67 m? 10.A car accelerates uniformly in a straight line from rest at the rate of 2.3 m/s 2. a.What is the speed of the car after it has traveled 55 m? b.How long does it take the car to travel 55 m?

© Houghton Mifflin Harcourt Publishing Company Example Problems Continued 11. A motorboat accelerates uniformly from a velocity of 6.5 m/s to the west to a velocity of 1.5 m/s to the west. If its acceleration was 2.7 m/s 2 to the east, how far did it travel during the acceleration? 12. An aircraft has a liftoff speed of 33 m/s. What minimum constant acceleration does this require if the aircraft is to be airborne after a takeoff run of 240 m? 13.A certain car is capable of accelerating at a uniform rate of 0.85 m/s 2. What is the magnitude of the car’s displacement as it accelerates uniformly from a speed of 83 km/h to one of 94 km/h?

© Houghton Mifflin Harcourt Publishing Company Homework Formative Assessment pg. 55 # 1-6

© Houghton Mifflin Harcourt Publishing Company Section 3 Falling Objects Preview Objectives Free Fall Free-Fall Acceleration Sample Problem Chapter 2

© Houghton Mifflin Harcourt Publishing Company Section 3 Falling Objects Chapter 2 Objectives Relate the motion of a freely falling body to motion with constant acceleration. Calculate displacement, velocity, and time at various points in the motion of a freely falling object. Compare the motions of different objects in free fall.

© Houghton Mifflin Harcourt Publishing Company Click below to watch the Visual Concept. Visual Concept Chapter 2 Section 3 Falling Objects Free Fall

© Houghton Mifflin Harcourt Publishing Company Chapter 2 Free Fall Free fall is the motion of a body when only the force due to gravity is acting on the body. The acceleration on an object in free fall is called the acceleration due to gravity, or free-fall acceleration. Free-fall acceleration is denoted with the symbols a g (generally) or g (on Earth’s surface). –g = m/s 2 on Earth (constant) Section 3 Falling Objects

© Houghton Mifflin Harcourt Publishing Company Click below to watch the Visual Concept. Visual Concept Chapter 2 Section 3 Falling Objects Free-Fall Acceleration

© Houghton Mifflin Harcourt Publishing Company Chapter 2 Free-Fall Acceleration Free-fall acceleration is the same for all objects, regardless of mass. This book will use the value g = 9.81 m/s 2. Free-fall acceleration on Earth’s surface is –9.81 m/s 2 at all points in the object’s motion. Consider a ball thrown up into the air. –Moving upward: velocity is decreasing, acceleration is – 9.81 m/s 2 –Top of path: velocity is zero, acceleration is –9.81 m/s 2 –Moving downward: velocity is increasing, acceleration is – 9.81 m/s 2 Section 3 Falling Objects

© Houghton Mifflin Harcourt Publishing Company Click below to watch the Visual Concept. Visual Concept Chapter 2 Section 3 Falling Objects Velocity and Acceleration of an Object in Free Fall

© Houghton Mifflin Harcourt Publishing Company Sample Problem Falling Object Jason hits a volleyball so that it moves with an initial velocity of 6.0 m/s straight upward. If the volleyball starts from 2.0 m above the floor, how long will it be in the air before it strikes the floor? Chapter 2 Section 3 Falling Objects

© Houghton Mifflin Harcourt Publishing Company Sample Problem, continued 1. Define Given:Unknown: v i = +6.0 m/s  t = ? a = –g = –9.81 m/s 2  y = –2.0 m Diagram: Place the origin at the Starting point of the ball (y i = 0 at t i = 0). Chapter 2 Section 3 Falling Objects

© Houghton Mifflin Harcourt Publishing Company Chapter 2 Sample Problem, continued 2. Plan Choose an equation or situation: Both ∆t and v f are unknown. Therefore, first solve for v f using the equation that does not require time. Then, the equation for v f that does involve time can be used to solve for ∆t. Rearrange the equation to isolate the unknown: Take the square root of the first equation to isolate v f. The second equation must be rearranged to solve for ∆t. Section 3 Falling Objects

© Houghton Mifflin Harcourt Publishing Company Chapter 2 Sample Problem, continued Tip: When you take the square root to find v f, select the negative answer because the ball will be moving toward the floor, in the negative direction. 3. Calculate Substitute the values into the equation and solve: First find the velocity of the ball at the moment that it hits the floor. Section 3 Falling Objects

© Houghton Mifflin Harcourt Publishing Company Chapter 2 Sample Problem, continued 4. Evaluate The solution, 1.50 s, is a reasonable amount of time for the ball to be in the air. Next, use this value of v f in the second equation to solve for ∆t. Section 3 Falling Objects

© Houghton Mifflin Harcourt Publishing Company Example Problem 1.A ball is thrown straight up into the air at an initial velocity of 25.0 m/s. Create a table showing the ball’s position, velocity, and acceleration each second for the first 5.00 seconds of its motion. 2.A robot probe drops a camera off the rim of a 239 m high cliff on Mars, where the free-fall acceleration is -3.7 m/s 2. a.Find the velocity with which the camera hits the ground. b.Find the time required for it to hit the ground.

© Houghton Mifflin Harcourt Publishing Company Example Problem Continued 3.A flowerpot falls from a windowsill 25.0 m above the sidewalk. a.How fast is the flowerpot moving when it strikes the ground? b.How much time does a passerby on the sidewalk below have to move out of the way before the flowerpot hits the ground? 4.A tennis ball is thrown vertically upward with an initial velocity of 8.0 m/s. a.What will the ball’s speed when it returns to its starting point? b.How long will the ball take to reach its starting point?

© Houghton Mifflin Harcourt Publishing Company Example Problem Continued 5.Jason hits a volleyball so that it moves with an initial velocity of 6.0 m/s straight upward. If the volleyball starts from 2.0 m above the floor, calculate its displacement when the volleyball’s final velocity is 1.1 m/s upward.

© Houghton Mifflin Harcourt Publishing Company Homework Formative Assessment pg. 61 #1-6