Chapter 2 Motion in One Dimension

 Classical Physics  Refers to physics before the 1900’s ▪ Kinematics/Dynamics ▪ Electromagnetism ▪ Thermodynamics  Applies to everyday phenomena  Modern Physics  Refers to post 1900’s physics ▪ Quantum Mechanics ▪ Relativity ▪ Nuclear Physics  Applies to very small/big phenomena

 Kinematics (special branch of mechanics)  Study of motion  Irrespective of causes (dynamics)  Three most important concepts ▪ Displacement ▪ Velocity ▪ Acceleration

 Assumptions  Ideal Particle ▪ Classical physics concept ▪ Point-like object, no size ▪ Real particles have size, charge, spin  Time is absolute ▪ Independent of position or velocity ▪ Relativity says time is not absolute! ▪ Twin Paradox

 Define frame of reference  Coordinate system ▪ Rectangular  Simplest case ▪ 1-Dimensional ▪ Can extrapolate to other dimensions (independent)  Only motion along the straight line is possible X

 Allows keep track of direction  Positive direction  Negative direction  When you apply equations for the motion of bodies it is very important to keep track of the direction of motion  Negative and Positive values  Affect result

The displacement of a moving object moving along the x-axis is defined as the change in the position of the object, Δx = x f – x i Where x i is the initial position and x f is the final position

First Displacement Δx 1 = x f – x i = 52 – 30 = 22 m Second Displacement Δx 2 = x f – x i = 38 – 52 = -14 m can have negative displacement – backwards

 Vector – both magnitude and direction  Ex: Velocity – magnitude (speed) and direction ▪ 55 mi/hr North East ▪ 45º North of East  Ex: Electric Field – magnitude and direction ▪ Electron ▪ Perpendicular to surface  Scalar – magnitude only  Ex: temperature, density, mass, volume NE E N

 Think of a vector as an arrow (pointing to some direction), and its magnitude as the length of the arrow (always positive, independent of the direction of the arrow).

You and your friend go for a walk to the park. On the way, your friend decides to text while walking and wanders off and takes a few side trips by dodging cars and falling off a bridge. When you both arrive at the park, do you and your friend have the same displacement? 1) yes 2) no ConcepTest Texting While Walking

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? 1) yes 2) no Yes, you have the same displacement. Since you and your friend had the same initial position and the same final position, then you have (by definition) the same displacement. ConcepTest Texting While Walking Follow-up: Have you and your friend traveled the same distance?

 The average speed of an object is given by: Average speed = total distance / total time s = d / t > 0 always  Speeds (m/s): ▪ Light 3 x 10 8 ▪ Sound 343 ▪ Person 10 ▪ Fastest Car 110 ▪ Continent 10 -8

 Speed and motion are relative  Depends on frame of reference Sun Earth Orbit speed of earth around the sun km/s A person running on earth - 5 m/s

 The average velocity v during a time interval Δt is the displacement Δx divided by Δt : v = — = ——  One dimension, straight line motion  The average velocity is equal to the slope of the straight line joining the initial and final points on a graph of the position vs. time t f - t i x f - x i ΔtΔt ΔxΔx (m/s)

 The figure below shows the graphical interpretation of the average velocity (A to B), in the case of an object moving with a variable velocity:

 Example: A motorist drives north for 35.0 minutes at 85.0 km/h and then stops for 15.0 minutes. He then continues north, traveling 130 km in 2.00 h. (a) What is his total displacement? (b) What is his average velocity?

 The slope of the line tangent to the position vs. time curve at some point is equal to the instantaneous velocity at that time. t x v = lim —— ΔxΔx ΔtΔt Δx -> 0

Position vs. Time graphs An object moving with a constant velocity will have a graph that is a straight line An object moving with a non-constant velocity will have a graph that is a curved line

 Position vs. Time  Time is always moving forward

Example: a)What is the velocity from O to A? b)What is the velocity from A to B? c)What is the velocity from O to C? d)What is the instantaneous velocity at t=2?

 Note: Average velocity does not necessarily have the same magnitude as average speed Average speed = ———————— Average velocity = ———————— Distance Travelled Time Displacement x1 x3 t1t2 s 12 = —————— ≠ 0 v 12 = —————— = 0 x2 x2 - x1 t2 - t1 x2 - x1 t2 - t1

Discussion Questions 1. A yellow car is heading East at 100 km/h and a red car is going North at 100 km/h. Do they have the same speed? Do they have the same velocity? 2. A 16-lb bowling ball in a bowling alley in Folsom Lanes heads due north at 10 m/s. At the same time, a purple 8-lb ball heads due north at 10 m/s in an alley in San Francisco. Do they have the same velocity?

The average acceleration v of an object with a change of velocity Δv during a time interval Δt : a = — = —— The instantaneous acceleration of an object at a certain time equals the slope of a velocity vs. time graph at that instant. t f - t i v f - v i ΔtΔt ΔvΔv (m/s 2 )

Average Acceleration

 Acceleration is a vector quantity  It has both a magnitude and a direction.  Positive or negative to indicate direction (of acceleration!)  Acceleration is positive when the velocity increases in the positive direction  Furthermore, the velocity increases when the acceleration and the velocity point in the same direction, it decreases when they are pointing in opposite directions

 Positive Acceleration  Negative Acceleration

 Positive Acceleration (Negative direction)

If the velocity of a car is non- zero (v  0), can the acceleration of the car be zero? 1) yes 2)no 3)depends on the velocity

constantvelocity zeroacceleration 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. If the velocity of a car is non- zero (v  0), can the acceleration of the car be zero? 1) yes 2)no 3)depends on the velocity

Velocity vs. Time graphs

 Example: A car traveling in a straight line has a velocity of m/s at some instant. After 4.0 s, its velocity is m/s. What is the car’s average acceleration during the 4.0-s time interval?

 Useful to look at situations when  the acceleration is constant ▪ Velocity is changing  Motion in a straight line  If the acceleration is constant, then the average acceleration is equal to the acceleration itself:  a = constant => = a = constant  Applies to gravity and other situations as well

 Important (and useful) equations with straight line, uniform acceleration 1. v = v 0 + at 2. Δx = vt = ½(v 0 + v)t 3. Δx = v 0 t + ½ at 2 4. v 2 = v a Δx  ALL you need to solve problems of kinematics with constant acceleration in 1 dimension

 Examples

 The most important example of 1d motion with uniform acceleration is gravity  A free-falling object is an object falling under the influence of gravity alone

 All objects fall near the earth’s surface with a constant acceleration, g  g = 9.8 m/s 2  g is always directed downward  All objects, regardless of mass, free-fall at the same acceleration

 An example of free fall is the collapse of gas in the formation of stars  Gravitational Force  Dependent on where we are on Earth, because the Earth is not a perfect sphere and because the presence of large masses (e.g. mountains) also affects the local gravitational force  For now, just consider the force of gravity as a constant force pulling any object towards the center of the Earth, and therefore perpendicular to the ground and towards the ground. ▪ Approximate constant acceleration pointing to the ground

 Feather and Hammer   Falling objects 

ConcepTest 2.9b Free Fall II 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? 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 v0v0v0v0 BillAlice vAvAvAvA vBvBvBvB

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. ConcepTest 2.9b Free Fall II 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? 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 v0v0v0v0 BillAlice vAvAvAvA vBvBvBvB Follow-up: Which one has the greater velocity when they hit the ground?

 All objects, regardless of mass, free-fall at the same acceleration  More detail in chapters on force and mass  Free-falling objects do not encounter air resistance  We can use the same four important equations from before (because g is constant acceleration) but change x direction to y direction  Substitute a = -g

Substitute a = -g, y-direction 1. v = v 0 - gt 2. Δy = vt = ½(v 0 + v)t 3. Δy = v 0 t - ½ gt 2 4. v 2 = v g Δy

Doing the Numbers  Imagine dropping an object, and measuring how fast it’s moving over consecutive 1 second intervals   The vertical component of velocity is changing by 9.8 m/s in each second, downwards  Let’s approximate this acceleration as 10 m/s 2

Time Interval Acceleration (m/s 2 down) Vel. at end of interval (m/s down) 0 – 1 s10 1 – 2 s – 3 s – 4 s – 5 s1050 Starting from rest, then letting go. After an interval t, the velocity changes by an amount at, so that v final = v initial + at How fast was it going at the end of 3 sec? v initial was 20 m/s after 2 sec a was 10 m/s (as always) t was 1 sec (interval) v final = 20 m/s + 10 m/s 2  1 s = 30 m/s

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. Then it accelerates downward due to gravity (again). 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. 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

a)Find the time when the stone reaches its maximum height. b)Determine the stone’s maximum height. c)Find the time the stone takes to return to its final position and find the velocity of the stone at that time. d)Find the time required for the stone to reach the ground. Example