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1 Unit 1: Kinematics 1.1 Scalars and Vectors in One Dimension 2 1.1.01 Scalar and Vector Quantities 2 1.1.02 Adding Vectors in One Dimension 3 1.2 Displacement, Velocity, and Acceleration 5 1.2.01 Definition of Displacement 5 1.2.02 Average Velocity and Acceleration 6 1.2.03 Instantaneous Velocity and Acceleration 8 1.3 Representing Motion 12 1.3.01 Constant Accelerated Motion in One Dimension 12 1.3.02 Free-Fall Kinematics 16 1.3.03 Representations of Motion in One Dimension 19 1.4 Reference Frames and Relative Motion 26 1.4.01 Reference Frames and Motion Variables 26 1.5 Vectors and Motion in Two Dimensions 29 1.5.01 Vector Quantities in Two Dimensions 29 1.5.02 Constant Accelerated Motion in Two Dimensions 32

Unit 1: Kinematics 2 Topic 1.1 Scalars and Vectors in One Dimension Learning Objectives • Understand that certain physical quantities can be represented by scalars or vectors. • Describe a scalar or vector quantity using magnitude and direction, as appropriate. • Understand how arrows can be used to represent vector values. • Learn that the addition of two physical quantities can represent other important quantities and can be determined by vector addition. Topic Questions • What is a scalar quantity? What is a vector quantity? • How does a vector represent a physically significant quantity with magnitude and direction? • How can the magnitude of a resultant vector be found when vectors are added together in one dimension? • How should vectors be drawn when added together in one dimension, and what is the direction of the resultant vector? 1.1.01 Scalar and Vector Quantities [ 1.1.A.1 1.1.A.2 ] When a physical quantity is measured experimentally, it is expressed by using SI units (or combinations of SI units). These quantities can be organized into two main categories: scalars, which are described by only a numerical value (eg, distance, speed), or vectors, which also include a direction (eg, displacement, velocity, acceleration). Scalar values are said to contain only a magnitude (ie, the numerical value), whereas vectors contain both magnitude and direction. To distinguish variables representing vector quantities, an arrow is placed above the variable: ⃗ = vectorquantity Visually, vectors can be represented by an arrow. The length of the arrow represents the vector's relative magnitude, and the direction of the arrow indicates the direction of the vector (see Figure 1.1). Figure 1.1 Vectors with different magnitudes and directions. The basic kinematic variables are displacement, velocity, and acceleration (see Topic 1.2). These variables are vector quantities because an object can move in any arbitrary direction in space. Vectors can also be assigned at specific points in space (in what are called vector fields) to represent the magnitude and direction of a physical quantity (eg, force) at each point in space.

Unit 1: Kinematics 3 1.1.02 Adding Vectors in One Dimension [ 1.1.A.3 1.1.B.1] In some cases, it may be necessary to consider the addition of two vectors, such as when two forces act on an object. The sum of two vectors in one dimension depends on whether the vectors are in the same direction or in opposite directions. As shown in Figure 1.2, when two vectors with magnitudes of vA and vB are in the same direction, the magnitude of their sum equals the sum of the magnitudes: ⃗!+ ⃗"= !+ "Also, the direction of their sum is the same as the direction of the original vectors. Figure 1.2 Adding two vectors with the same direction in one dimension. However, Figure 1.3 shows that when two vectors of magnitude vA and vC are in opposite directions, the magnitude of their sum equals the difference of the magnitudes: ⃗!+ ⃗#= !− #In addition, the direction of their sum is the same as the direction of the original vector with the largest magnitude. Furthermore, the sum of vectors ⃗!, ⃗", and ⃗# is equal to: ⃗!+ ⃗"+ ⃗#= !+ "− # Figure 1.3 Adding vectors with opposite directions in one dimension.

Unit 1: Kinematics 4 Topic 1.1 Scalars and Vectors in One Dimension Check for Understanding Quiz 1. Vector A ""⃗ is shown below. Which of the following arrows best represents vector B if ""⃗=""⃗? A. B. C. D. 2. Vector → and → are shown below. If vectors →+ →+ →= , which arrow best represents vector →? A. B. C. D. Note: Answers to this quiz are in the back of the book (appendix).

Unit 1: Kinematics 5 Topic 1.2 Displacement, Velocity, and Acceleration Learning Objectives • Determine the change in an object's position. • Describe the average velocity and acceleration of an object. • Describe the instantaneous velocity and acceleration of an object. Topic Questions • What are the definitions of displacement, velocity, and acceleration? • How can the change in position, velocity, or acceleration of an object be determined? • How do instantaneous and average values of velocity or acceleration differ? 1.2.01 Definition of Displacement [ 1.2.A.1 1.2.A.2 ] The motion of an object is described by its position x, velocity v, and acceleration a. These variables are vector quantities (ie, they have both magnitude and direction). In this section, we describe how these variables are used in problems on the exam, focusing particularly on motion in one dimension. Distance and Displacement Distance is a scalar quantity defined as the total path length traveled by an object or system in question. However, displacement Δ⃗ is a vector quantity equal to the difference between the final position ⃗! and the initial position ⃗" of an object: Δ⃗ = ⃗!− ⃗" For example, consider a student who walks forward and backward along a straight line while a motion detector is used to produce the graph of position as a function of time shown in Figure 1.4. Figure 1.4 Calculating distance and displacement.

Unit 1: Kinematics 6 The total distance the student walked is equal to the sum of the distances traveled for t = 0 s to t = 1 s, t = 2 s to t = 3 s, and t = 4 s to t = 5 s (note that the distances traveled during the intervals of t = 1 s to t = 2 s and t = 3 s to t = 4 s are equal to 0 m): distance = 2m + 1m + 2m = 5m However, the displacement from the starting location is calculated only from the final position, 3 m, and the initial position, 4 m: Δ = 3m−4m = −1m Therefore, the student traveled a total distance of 5 m but had an overall displacement of only 1 m in the negative direction. 1.2.02 Average Velocity and Acceleration [ 1.2.B.1 1.2.B.2 1.2.B.3 1.2.B.4 ] The velocity of an object measures its change in position over time. Furthermore, the acceleration of an object is defined as any change in its velocity (speed or direction) over time. Average values for an object's velocity and acceleration can be calculated to describe its motion. Average Velocity Speed and velocity both describe how fast an object is moving. However, velocity is a vector value that also describes the direction of its motion. An object's average velocity →#$% is a vector defined as its change in position Δ → (ie, displacement) over an interval Δt: →#$%=Δ →Δ In terms of final and initial positions and times: ⃗#$%=⃗!− ⃗"!− " For example, a toy car moves back and forth in one dimension over a total interval of 10 s. The toy car moves 4 m forward, then 9 m backward, and finally another 2 m forward. Assuming that "forward" is the positive direction, the total displacement of the toy car over the 10 s interval is equal to: Δ →=(4m)+(−9m)+(2m)=−3m Hence, the average velocity of the car is equal to: →#$%=Δ →Δ=−3m10s= −0.3ms The average velocity of the car indicates an overall negative direction to the car's motion. Average Acceleration The average acceleration ⃗#$% of an object measures the change in velocity Δ⃗ over a finite interval Δt: ⃗#$%=Δ⃗Δ In terms of final and initial velocities and times: ⃗#$%=⃗!− ⃗"!− "

Unit 1: Kinematics 7 For example, the velocity of a car moving in one dimension is plotted over time in the graph shown in Figure 1.5. The average acceleration of the car can be determined by analyzing the change in velocity during the 8 s interval. Figure 1.5 Average acceleration from a velocity versus time graph. The car has an initial velocity vi = 0 m/s at ti = 0 s and final velocity vf = 4 m/s at tf = 8 s. Therefore, inserting the values in the average acceleration equation for the car yields: =ΔΔ=?4ms@−?0ms@(8s − 0s) =4ms8s= 0.5ms& Therefore, the average acceleration of the car is equal to 0.5 m/s2.

Unit 1: Kinematics 8 1.2.03 Instantaneous Velocity and Acceleration [ 1.2.B.5 ] The average values of an object's velocity or acceleration only consider the initial and final points of its motion. Alternatively, the instantaneous velocity and acceleration of an object can provide a more precise way of describing its motion over very small intervals. Instantaneous Velocity The velocity → of an object is equal to the change in its position Δ⃗ (ie, displacement) over an interval Δt: →=Δ⃗Δ The instantaneous velocity of an object is the velocity at a specific instant in time determined by considering Δ⃗ over an infinitesimal interval of motion (ie, as Δt approaches 0) around a given time: →= lim'(→*Δ →Δ Figure 1.6 illustrates that the slope of the tangent line at each point on a position versus time graph is equal to the instantaneous velocity for that instant in time. Figure 1.6 Instantaneous velocity from a position versus time graph. For instance, consider the position versus time graph of an object as shown in Figure 1.7. Because the total displacement of the object from 0 s to 4 s is equal to zero, its average velocity over that interval is equal to zero. However, a more accurate description of the object's motion would be to estimate the instantaneous velocities at the points labeled A, B, and C.

Unit 1: Kinematics 9 Figure 1.7 Ranking velocity of an object at different times. The instantaneous velocity at a specific point is equal to the slope of the tangent line at that point. The slope of the graph has the same magnitude at location A and C, but the slope is positive at point A and negative at point C. Therefore, the object has the same speed at points A and C but is traveling in opposite directions. += −, Furthermore, the slope at point B is equal to zero. Therefore, the velocity at point B is also equal to zero. -= 0 Instantaneous Acceleration The acceleration → of an object is equal to the change in the object's velocity Δ → over an interval Δt: →=Δ →Δ Moreover, the instantaneous acceleration at a specific time follows from considering Δ → over an infinitesimal Δt around that time: →= lim'(→*Δ →Δ Consequently, the instantaneous acceleration can be determined by calculating the slope of a line tangent to the velocity curve at an instant, as shown in Figure 1.8. Figure 1.8 Instantaneous acceleration from a velocity versus time graph.

Unit 1: Kinematics 10 For example, a group of students record the velocity of a cart on a horizontal track and produce the graph shown in Figure 1.9. Figure 1.9 Calculating instantaneous acceleration from a velocity versus time graph. In this example, the overall change in the velocity of the cart during the entire 4 s interval is equal to zero. Therefore, the average acceleration of the cart is also equal to zero. Calculating the instantaneous accelerations at specific times provides a more accurate description of the object's acceleration. The cart's velocity is constant or changing linearly over each 1 s interval. Therefore, the cart's instantaneous acceleration is equal to the slope of the graph over each interval. From t = 0 s to 1 s, the acceleration is zero because the velocity is constant: */01= 0ms& From 1 s to 2 s, the acceleration is negative: 1/0&=ΔΔ=0ms− 1.5ms1s= −1.5ms& From 2 s to 3 s, the acceleration is positive: &/02=ΔΔ=1.5ms− 0ms1s= 1.5ms& Finally, from 3 s to 4 s, the acceleration is again zero because the slope of the graph is zero: 2/03= 0ms&

Unit 1: Kinematics 11 Topic 1.2 Displacement, Velocity, and Acceleration Check for Understanding Quiz 1. A person rides a bicycle back and forth along a road. He travels 4 m west, then turns around and travels 6 m east, and then turns back around and travels another 10 m west. What is the displacement of the person if the positive direction is to the east? A. +8 m B. −8 m C. +20 m D. −20 m 2. The position versus time graph of an object is shown below. What is the instantaneous velocity of the object at t = 2 s? A. +0.75 m/s B. −0.75 m/s C. +1.5 m/s D. −1.5 m/s 3. A car traveling to the left with an initial speed of 20 m/s comes to a full stop in a time of 5 s. Which statement best describes the acceleration of the car if the positive direction is to the right? A. The acceleration is positive. B. The acceleration is negative. C. The acceleration points to the left. D. The acceleration is zero. Note: Answers to this quiz are in the back of the book (appendix).