1-Dimensional Motion

Topics of Physics Physics Mechanics Kinematics Dynamics Vibrations and Waves Optics Electro-magnetism Thermo-dynamics Relativity Quantum

Mechanics How Fast? How Far? How Long? Why? Mechanics Kinematics Dynamics Kinematics The study of how things move How far? How Fast? How long? Dynamics The study of why things move Force causes motion and changes in motion We are focusing on Kinematics for now. We are learning how to describe the motion of objects. We will study Dynamics later in the term. How Fast? How Far? How Long? Why?

One Dimensional Motion To simplify the concept of motion, we will first consider motion that takes place in one dimension. One example is the motion of a commuter train on a straight track. The train can either move forward or backward along a straight line. To describe motion, you must first choose a frame of reference. A frame of reference is a system for specifying the precise location of objects in space and time. The frame of reference for one dimensional motion can be thought of as a number line. Without a frame of reference, it would be impossible to meaningfully describe the motion of any object.

ILD – Kinematics 1 – Human Motion Demonstration 1 Moving Away Moving Toward time distance time distance

ILD – Kinematics 1 – Human Motion Demonstration 2 Moving Away Moving Toward time velocity time velocity

ILD – Kinematics 1 – Human Motion Demonstration 3 Moving Away at twice the speed time distance time velocity

ILD – Kinematics 1 – Human Motion Demonstration 4 Time (s) velocity Time (s) distance

Frame of Reference Are you at rest? Just exactly how fast are you and I moving?

Displacement Concept Check You and your dog go for a walk to the park. On the way, your dog takes many side trips to chase squirrels or examine fire hydrants. When you arrive at the park, do you and your dog have the same displacement? Yes No

Displacement Concept Check You and your dog go for a walk to the park. On the way, your dog takes many side trips to chase squirrels or examine fire hydrants. When you arrive at the park, do you and your dog have the same displacement? Yes No Yes, you have the same displacement. Since you and your dog had the same initial position and the same final position, then you have (by definition) the same displacement. Follow-up: Have you and your dog traveled the same distance?

Displacement Concept Check Does the displacement of an object depend on the specific location of the origin of the coordinate system? Yes No It depends on the coordinate system 10 20 30 40 50 10 20 30 40 50 30 40 50 60 70 30 40 50 60 70

Displacement Concept Check Does the displacement of an object depend on the specific location of the origin of the coordinate system? Yes No It depends on the coordinate system 10 20 30 40 50 10 20 30 40 50 30 40 50 60 70 30 40 50 60 70 Since the displacement is the difference between two coordinates, the origin does not matter.

Displacement Concept Check Does the odometer in a car measure distance or displacement? Distance Displacement Both

Displacement Concept Check Does the odometer in a car measure distance or displacement? Distance Displacement Both If you go on a long trip and then return home, your odometer does not measure zero, but it records the total miles that you traveled. That means the odometer records distance. Follow-up: How would you measure displacement in your car?

displacement = change in position = final position – initial position Displacement = How Far Displacement ≠ Distance Displacement compares start point to end point Distance = Only How Far Displacement = How Far AND In What Direction Displacement is a change in position. Displacement is not always equal to the distance traveled. If the gecko were to turn around and return to its original position, its displacement would be zero, while its distance traveled would be something like 120 cm. The magnitude of displacement is equal to distance traveled only if there is no change in direction. The SI unit of displacement is the meter, m. Dx = xf – xi displacement = change in position = final position – initial position

displacement = change in position = final position – initial position Displacement = How Far yf Displacement is a change in position We normally use the variable x to represent horizontal positions and displacements and the variable y to represent vertical positions and displacements. The SI unit of displacement is the meter, m. yi Dy = yf – yi displacement = change in position = final position – initial position

Positive and Negative Displacements Sign indicates direction

Speed Concept Check If the position of a car is zero, does its speed have to be zero? Yes No It depends on the origin -10 10

Speed Concept Check If the position of a car is zero, does its speed have to be zero? Yes No It depends on the origin -10 10 No, the speed does not depend on position, it depends on the change of position. Since we know that the displacement does not depend on the origin of the coordinate system, an object can easily start at x = –3 and be moving by the time it gets to x = 0.

Speed Concept Check Does the speedometer in a car measure velocity or speed? Velocity Speed Both Neither

Speed Concept Check Does the speedometer in a car measure velocity or speed? Velocity Speed Both Neither The speedometer clearly measures speed, not velocity. Velocity depends on direction (vector), but the speedometer does not care what direction you are traveling. It only measures the magnitude of the velocity, which is the speed. Follow-up: How would you measure velocity in your car?

Velocity = How Fast xf xi xi xf -10 10 Velocity ≠ Speed Speed = How Fast Velocity = How Fast AND In What Direction Sign indicates direction xf xi xi xf -10 10 If a car is driving along a road and is at position xi at some time and position xf at some later time, the average velocity of the car can be determined by the change in position divided by the time it took to make that change. Of course, if the car turned around and traveled in the opposite direction, the change in position (displacement) would be negative, and so would the average velocity.

Vectors v. Scalars Vectors have magnitude and direction Displacement Velocity Acceleration Scalars have only magnitude Distance Speed Mass Time Volume Vector: I'm applying for a villain loan. I go by the name of Vector. It's a mathematical term, represented by an arrow with both direction and magnitude. Vector! That's me, because I commit crimes with both direction and magnitude. Oh yeah!  We will learn more about Vectors next unit. For now, we will represent the direction of a vector with a (+) or (-) sign.

Graphing Velocity xf xi xi xf 15 position (m) 10 5 6 12 time (s) If a car is driving along a road and is at position xi at some time and position xf at some later time, the average velocity of the car can be determined by the change in position divided by the time it took to make that change. Of course, if the car turned around and traveled in the opposite direction, the change in position (displacement) would be negative, and so would the average velocity. 10 5 6 12 time (s)

Graphing Velocity Table 2 Position-Time Data t (s) x (m) 0.0 1.0 2.0 8.0 3.0 18.0 4.0 32.0 30 Position (m) 20 10 The instantaneous velocity is the velocity of an object at some instant or at a specific point in the object’s path. If the position-time graph of an object in motion is curved like this, the object is not traveling at a constant velocity – its velocity is continuously changing. The instantaneous velocity at any time can be determined by measuring the slope of the line that is tangent to that point on the position-time graph. 1.0 2.0 3.0 4.0 Time (s)

Summarize What are the key ideas from this lesson? What connections can I make with other ideas? What questions do I still have?

GUESS Method Given Unknown Equation(s) Substitute Significant Figures State the relevant given information in variable form, w/ units. Include a diagram when appropriate. Unknown State the unknown variable (that which you are attempting to find). Equation(s) State the appropriate equation(s). Solve for the unknown if not already. Substitute Substitute the given information into the appropriate equation(s). Include units with all values. Significant Figures Round the answer to the appropriate number of significant figures based on the given information. Solve the problem on the template provided, following the GUESS Method.

Sample Problem A speeder passes a parked police car at 26 m/s. The police car starts from rest with a uniform acceleration of 2.44 m/s2. How much time passes before the police car achieves the same speed as the speeding car? Solve the problem on the template provided, following the GUESS Method.

Solution 1. Define Given: vi = 0.0 m/s (has ∞ sig. figs.) vf = 26 m/s (2 sig. figures) a = 2.44 m/s2 (3 sig. figures) Unknown: t = ? Choose a coordinate system. The most convenient one has an origin at the parked police car. The positive direction is in the direction the speeding car is traveling. Solve the problem on the template provided, following the GUESS Method.

Solution 2. Plan Equation: Because the initial and final velocities and the acceleration is known, the elapsed time can be found using the following equation: Since the equation is not already solved for the unknown, rearrange the equation to isolate the unknown. In this case, subtract vi from both sides first, then divide by a. Solve the problem on the template provided, following the GUESS Method.

Solution 3. Calculate Substitute: Remember to include units with your values. Significant Digits: Since the fewest number of significant figures among given information is 2, the answer must be rounded to 2 significant figures. Solve the problem on the template provided, following the GUESS Method.

Solution 4. Evaluate An acceleration of 2.44 m/s2 means that the police car will change its speed by 2.44 m/s each second. Starting from rest, it will be moving at 2.44 m/s after 1 second, 4.88 m/s after 2 seconds, and 7.32 m/s after 3 seconds and so on. After 10 seconds, the car will reach a speed of 24.4 m/s, so it is reasonable that after approximately 11 seconds, the police car will reach a speed of approximately 26 m/s. Solve the problem on the template provided, following the GUESS Method.

Problems Sample

Problems Sample

Problems – 2 𝑣 𝑎𝑣𝑔 =48.0 km h ∆𝑥=144 km ∆𝑡=? 𝑣 𝑎𝑣𝑔 = ∆𝑥 ∆𝑡 → ∆𝑡= ∆𝑥 𝑣 𝑎𝑣𝑔 ∆𝑡= ∆𝑥 𝑣 𝑎𝑣𝑔 = 144 km 48.0 km h =3.00 ℎ

Problems – 4

Graphing Velocity Table 2 Position-Time Data t (s) x (m) 0.0 1.0 2.0 8.0 3.0 18.0 4.0 32.0 30 Position (m) 20 10 The instantaneous velocity is the velocity of an object at some instant or at a specific point in the object’s path. If the position-time graph of an object in motion is curved like this, the object is not traveling at a constant velocity – its velocity is continuously changing. The instantaneous velocity at any time can be determined by measuring the slope of the line that is tangent to that point on the position-time graph. 1.0 2.0 3.0 4.0 Time (s)

Acceleration Concept Check If the velocity of a car is non-zero (v ¹ 0), can the acceleration of the car be zero? Yes No Depends on velocity

Acceleration Concept Check If the velocity of a car is non-zero (v ¹ 0), can the acceleration of the car be zero? Yes No Depends on velocity Sure it can! An object moving with constant velocity has a non-zero velocity, but it has zero acceleration since the velocity is not changing.

Acceleration Acceleration ≠ How Fast Acceleration = How Fast Velocity changes Velocity = How Fast AND In What Direction Sign indicates how Velocity is changing 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.

Graphing Acceleration Speeding up while moving in the positive direction 15 Velocity (m/s) 10 5 If a car is driving along a road and is at position xi at some time and position xf at some later time, the average velocity of the car can be determined by the change in position divided by the time it took to make that change. Of course, if the car turned around and traveled in the opposite direction, the change in position (displacement) would be negative, and so would the average velocity. 1 5 time (s)

Speeding Up in Positive Direction

Speeding Up in Negative Direction

Slowing Down in Positive Direction

Slowing Down in Negative Direction

Position Graph Concept Check The graph of position versus time for a car is given below. What can you say about the velocity of the car over time? t x 1. it speeds up all the time 2. it slows down all the time 3. it moves at constant velocity 4. sometimes it speeds up and sometimes it slows down 5. not really sure

Position Graph Concept Check The graph of position versus time for a car is given below. What can you say about the velocity of the car over time? t x 1. it speeds up all the time 2. it slows down all the time 3. it moves at constant velocity 4. sometimes it speeds up and sometimes it slows down 5. not really sure The car moves at a constant velocity because the x vs. t plot shows a straight line. The slope of a straight line is constant. Remember that the slope of x versus t is the velocity.

Position Graph Concept Check The graph of position versus time for a car is given below. What can you say about the velocity of the car over time? t x 1. it speeds up all the time 2. it slows down all the time 3. it moves at constant velocity 4. sometimes it speeds up and sometimes it slows down 5. not really sure

Position Graph Concept Check The graph of position versus time for a car is given below. What can you say about the velocity of the car over time? 1. it speeds up all the time 2. it slows down all the time 3. it moves at constant velocity 4. sometimes it speeds up and sometimes it slows down 5. not really sure The car slows down all the time because the slope of the x vs. t graph is diminishing as time goes on. Remember that the slope of x vs. t is the velocity. At large t, the value of the position x does not change, indicating that the car must be at rest.

Velocity Graph Concept Check Consider the line labeled A in the v versus t plot. How does the speed change with time for line A? 1. decreases 2. increases 3. stays constant 4. increases, then decreases 5. decreases, then increases v t A

Velocity Graph Concept Check Consider the line labeled A in the v versus t plot. How does the speed change with time for line A? 1. decreases 2. increases 3. stays constant 4. increases, then decreases 5. decreases, then increases v t A In case A, the initial velocity is positive and the magnitude of the velocity continues to increase with time.

Velocity Graph Concept Check Consider the line labeled A in the v versus t plot. How does the speed change with time for line A? 1. decreases 2. increases 3. stays constant 4. increases, then decreases 5. decreases, then increases v t A

Velocity Graph Concept Check Consider the line labeled A in the v versus t plot. How does the speed change with time for line A? 1. decreases 2. increases 3. stays constant 4. increases, then decreases 5. decreases, then increases v t A

Velocity Graph Concept Check Consider the line labeled A in the v versus t plot. How does the speed change with time for line A? 1. decreases 2. increases 3. stays constant 4. increases, then decreases 5. decreases, then increases v t A

Velocity Graph Concept Check Consider the line labeled A in the v versus t plot. How does the speed change with time for line A? 1. decreases 2. increases 3. stays constant 4. increases, then decreases 5. decreases, then increases v t A

Velocity Graph Concept Check Consider the line labeled A in the v versus t plot. How does the speed change with time for line A? 1. decreases 2. increases 3. stays constant 4. increases, then decreases 5. decreases, then increases v t A

Velocity Graph Concept Check Consider the line labeled A in the v versus t plot. How does the speed change with time for line A? 1. decreases 2. increases 3. stays constant 4. increases, then decreases 5. decreases, then increases v t A

Velocity Graph Concept Check Consider the line labeled B in the v versus t plot. How does the speed change with time for line B? v t A B 1. decreases 2. increases 3. stays constant 4. increases, then decreases 5. decreases, then increases

Velocity Graph Concept Check Consider the line labeled B in the v versus t plot. How does the speed change with time for line B? v t A B 1. decreases 2. increases 3. stays constant 4. increases, then decreases 5. decreases, then increases In case B, the initial velocity is positive and the magnitude of the velocity decreases toward zero. This will bring the object to rest briefly. Then the velocity becomes negative which means the object has changed direction. Velocity becomes more negative which means it speeds up in the opposite direction.

Velocity and Acceleration P. 51

Equations for Motion Constant Velocity Acceleration (change in velocity) position (m) Velocity (m/s) time (s) time (s) slope= ∆𝑥 ∆𝑡 = 𝑣 𝑎𝑣𝑔 slope= ∆𝑣 ∆𝑡 = 𝑎 𝑎𝑣𝑔 𝑣 𝑎𝑣𝑔 = ∆𝑥 ∆𝑡 𝑎 𝑎𝑣𝑔 = ∆𝑣 ∆𝑡

Problems Sample A Shuttle bus speeds up with an average acceleration of +1.8m/s2. How long does it take the bus to speed up from rest to 9.0 m/s?

Problems – 8 8.

Acceleration, continued Velocity Time Describe the motion of the object whose velocity v. time graph looks like this. What would the acceleration v. time graph look like for this object? Acceleration Time

Acceleration, continued Position Slope (Derivative) Area (Integral) Time Velocity Time Slope (Derivative) Area (Integral) Describe the motion of the object whose velocity is as shown. Time Acceleration

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 or uniform acceleration. These equations use the following 5 variables: Dx = displacement vi = initial velocity vf = final velocity Dt = time interval a = acceleration

Equations for Constantly Accelerated Straight-Line Motion

Kinematics Equations a Dx vi vf Dt                

Problems Sample A racing car reaches a speed of 42 m/s. It then begins a uniform negative acceleration, using its parachute and braking system, and comes to rest 5.5 s later. Find the distance that the car travels during braking

Problems – 14

Law of Falling Bodies

Hammer and Feather

Hammer and Feather (2)

Equations for Free Fall (Constant Acceleration)

Problems – 21 21.

Free Fall Concept Check v t 1 v t 3 v t 4 v t 2 You toss a baseball straight up. For the upward part of the motion only, which of the above plots represents the v vs. t graph for this motion? (Assume the positive direction is up.)

Free Fall Concept Check v t 1 v t 3 v t 4 v t 2 You toss a baseball straight up. For the upward part of the motion only, which of the above plots represents the v vs. t graph for this motion? (Assume the positive direction is up.) The ball slows down as it moves in the positive direction because gravity pulls downwards on the ball as it moves up. Since the y-axis is pointing upwards and it accelerates downward, its velocity decreases to zero.

Free Fall Concept Check v t 1 1 v t 2 v t 3 v t 4 4 2 3 You toss a baseball straight up. Right after it reachest its highest point and before it is caught, which of the above plots represents the v vs. t graph for this motion? (Assume the positive direction is up.)

Free Fall Concept Check v t 1 1 v t 2 v t 3 v t 4 4 2 3 You toss a baseball straight up. Right after it reachest its highest point and before it is caught, which of the above plots represents the v vs. t graph for this motion? (Assume the positive direction is up.) At the top, the ball has a velocity of zero (v = 0). As it falls, its velocity is negative and it increases in speed as it falls. Therefore, velocity is negative and becomes more and more negative as it accelerates downward.

Free Fall

Velocity and Acceleration of an Object in Free Fall Position Time Velocity Time Describe the motion of the object whose velocity is as shown. Time Acceleration -9.81 m/s2

Concept Check – Free Fall You throw a ball straight up into the air. After it leaves your hand, at what point in its flight does it have the maximum value of acceleration? 1. its acceleration is constant everywhere 2. at the top of its trajectory 3. halfway to the top of its trajectory 4. just after it leaves your hand 5. just before it returns to your hand on the way down

Concept Check – Free Fall You throw a ball straight up into the air. After it leaves your hand, at what point in its flight does it have the maximum value of acceleration? 1. its acceleration is constant everywhere 2. at the top of its trajectory 3. halfway to the top of its trajectory 4. just after it leaves your hand 5. just before it returns to your hand on the way down The ball is in free fall once it is released. Therefore, it is entirely under the influence of gravity, and the only acceleration it experiences is g, which is constant at all points.

Concept Check – Free Fall Alice and Bill are at the top of a building. Alice throws her ball downward. Bill simply drops his ball. Which ball has the greater acceleration just after release? 1. Alice’s ball 2. it depends on how hard the ball was thrown 3. Neither – they both have the same acceleration 4. Bill’s ball v0 Bill Alice vA vB

Concept Check – Free Fall Alice and Bill are at the top of a building. Alice throws her ball downward. Bill simply drops his ball. Which ball has the greater acceleration just after release? 1. Alice’s ball 2. it depends on how hard the ball was thrown 3. Neither – they both have the same acceleration 4. Bill’s ball Both balls are in free fall once they are released, therefore they both feel the acceleration due to gravity (g). This acceleration is independent of the initial velocity of the ball. v0 Bill Alice vA vB Follow-up: Which one has the greater velocity when they hit the ground?

Concept Check – Free Fall You throw a ball upward with an initial speed of 10 m/s. Assuming that there is no air resistance, what is its speed when it returns to you? 1. more than 10 m/s 2. 10 m/s 3. less than 10 m/s 4. zero 5. need more information

Concept Check – Free Fall You throw a ball upward with an initial speed of 10 m/s. Assuming that there is no air resistance, what is its speed when it returns to you? 1. more than 10 m/s 2. 10 m/s 3. less than 10 m/s 4. zero 5. need more information The ball is slowing down on the way up due to gravity. Eventually it stops. Since a = -g on the way up AND on the way down, the ball reaches the same speed when it gets back to you as it had when it left.

Concept Check – Free Fall Alice and Bill are at the top of a cliff of height H. Both throw a ball with initial speed v0, Alice straight down and Bill straight up. The speeds of the balls when they hit the ground are vA and vB. If there is no air resistance, which is true? 1. vA < vB 2. vA = vB 3. vA > vB 4. impossible to tell v0 Bill Alice H vA vB v0 Bill Alice H vA vB [CORRECT 5 ANSWER]

Concept Check – Free Fall Alice and Bill are at the top of a cliff of height H. Both throw a ball with initial speed v0, Alice straight down and Bill straight up. The speeds of the balls when they hit the ground are vA and vB. If there is no air resistance, which is true? 1. vA < vB 2. vA = vB 3. vA > vB 4. impossible to tell v0 Bill Alice H vA vB Bill’s ball goes up and comes back down to Bill’s level. At that point, it is moving downward with v0, the same as Alice’s ball. Thus, it will hit the ground with the same speed as Alice’s ball. [CORRECT 5 ANSWER] Follow-up: What happens if there is air resistance?

Concept Check – Free Fall You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their velocities? 1. They both increase at the same rate 2. The velocity of the first rock increases faster than the second 3. The velocity of the second rock increases faster than the first 4. Both velocities remain constant

Concept Check – Free Fall You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their velocities? 1. They both increase at the same rate 2. The velocity of the first rock increases faster than the second 3. The velocity of the second rock increases faster than the first 4. Both velocities remain constant Both rocks are in free fall, thus under the influence of gravity only. That means they both experience the constant acceleration of gravity. Since acceleration is defined as the change of velocity, both of their velocities increase at the same rate. Follow-up: What happens when air resistance is present?

Concept Check – Free Fall You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their separation? 1. the separation increases as they fall 2. the separation stays constant at 4 m 3. the separation decreases as they fall 4. it is impossible to answer without more information

Concept Check – Free Fall You drop a rock off a bridge. When the rock has fallen 4 m, you drop a second rock. As the two rocks continue to fall, what happens to their separation? 1. the separation increases as they fall 2. the separation stays constant at 4 m 3. the separation decreases as they fall 4. it is impossible to answer without more information At any given time, the first rock always has a greater velocity than the second rock, therefore it will always be increasing its lead as it falls. Thus, the separation will increase.

Free Fall Concept Check When throwing a ball straight up, which of the following is true about its velocity v and its acceleration a at the highest point in its path? 1. both v = 0 and a = 0 2. v ¹ 0, but a = 0 3. v = 0, but a ¹ 0 4. both v ¹ 0 and a ¹ 0 5. not really sure

Free Fall Concept Check When throwing a ball straight up, which of the following is true about its velocity v and its acceleration a at the highest point in its path? 1. both v = 0 and a = 0 2. v ¹ 0, but a = 0 3. v = 0, but a ¹ 0 4. both v ¹ 0 and a ¹ 0 5. not really sure At the top, clearly v = 0 because the ball has momentarily stopped. But the velocity of the ball is changing, so its acceleration is definitely not zero. Otherwise it would remain at rest! Follow-up: …and the value of a is…?

Velocity Graph Concept Check (3) 1 v t 3 v t 4 v t 2 You drop a very bouncy rubber ball. It falls, and then it hits the floor and bounces right back up to you. Which of the following represents the v vs. t graph for this motion?

Velocity Graph Concept Check (3) 1 v t 3 v t 4 v t 2 You drop a rubber ball. Right after it leaves your hand and before it hits the floor, which of the above plots represents the v vs. t graph for this motion? (Assume your y-axis is pointing up.) Initially, the ball is falling down, so its velocity must be negative (if UP is positive). Its velocity is also increasing in magnitude as it falls. Once it bounces, it changes direction and then has a positive velocity, which is also decreasing as the ball moves upward.