2 2.1 Displacement & Velocity Learning Objectives Describe motion in terms of displacement, time, and velocityCalculate the displacement of an object traveling at a known velocity for a specific time intervalConstruct and interpret graphs of position versus time
3 Essential Concepts Frames of reference Vector vs. scalar quantities DisplacementVelocityAverage velocityInstantaneous velocityAccelerationGraphical representation of motion
4 Reference Frames Motion is relative When we say an object is moving, we mean it is moving relative to something else (reference frame)
5 Scalar Quantities & Vector Quantities Scalar quantities have magnitudeExample: speed 15 m/sVector quantities have magnitude and directionExample: velocity 15 m/s North
6 Displacement ∆x = xf - xi Displacement is a vector quantity Indicates change in location (position) of a body∆x = xf - xiIt is specified by a magnitude and a direction.Is independent of the path traveled by an object.
8 Displacement vs. Distance Distance is the length of the path that an object travelsDisplacement is the change in position of an object
9 Describing Motion Describing motion requires a frame of reference
10 Determining Displacement In these examples, position is determined with respect to the origin, displacement wrt x1
11 Indicating Direction of Displacement Direction can be indicated by sign, degrees, or geographical directions.When sign is used, it follows the conventions of a standard graphPositiveRightUpNegativeLeftDown
12 Displacement Linear change in position of an object Is not the same as distance
13 Displacement Distance = length (blue) How many units did the object move?Displacement = change in position (red)How could you calculate the magnitude of line AB?≈ 5.1 units, NE
14 Reference Frames & Displacement Direction is relative to the initial position, x1x1 is the reference point
15 Average VelocitySpeed: how far an object travels in a given time intervalVelocity includes directional information:
17 VelocityExampleA squirrel runs in a straight line, westerly direction from one tree to another, covering 55 meters in 32 seconds. Calculate the squirrel’s average velocityvavg = ∆x / ∆tvavg = 55 m / 32 svavg = 1.7 m/s west
18 Velocity can be represented graphically: Position Time Graphs
19 Velocity can be interpreted graphically: Position Time Graphs Find the average velocity between t = 3 min to t = 8 min
20 Calculate the average velocity for the entire trip
21 Formative Assessment: Position-Time Graphs Object at rest?Traveling slowly in a positive direction?Traveling in a negative direction?Traveling quickly in a positive direction?dev.physicslab.org
22 Average vs. Instantaneous Velocity Velocity at any given moment in time or at a specific point in the object’s path
27 2.2 Acceleration Learning Objectives Describe motion in terms of changing velocityCompare graphical representations of accelerated and non-accelerated motionsApply kinematic equations to calculate distance, time, or velocity under conditions of constant acceleration
29 AccelerationAcceleration is the rate of change of velocity.
30 Acceleration: Change in Velocity Acceleration is the rate of change of velocitya = ∆v/∆ta = (vf – vi) / (tf – ti)Since velocity is a vector quantity, velocity can change in magnitude or directionAcceleration occurs whenever there is a change in magnitude or direction of movement.
31 Acceleration Because acceleration is a vector, it must have direction Here is an example of negative acceleration:
32 Customary Dimensions of Acceleration a = ∆v/∆t= m/s/s= m/s2Sample problems 2BA bus traveling at 9.0 m/s slows down with an average acceleration of -1.8 m/s. How long does it take to come to a stop?
33 Negative Acceleration Both velocity & acceleration can have (+) and (-) valuesNegative acceleration does not always mean an object is slowing down
34 Is an object speeding up or slowing down? Depends upon the signs of both velocity and accelerationConstruct statement summarizing this table.VelocityAccelMotion+Speeding up in + dir-Speeding up in - dirSlowing down in + dirSlowing down in - dir
35 Velocity-Time Graphs Is this object accelerating? How do you know? What can you say about its motion?
36 Velocity-Time Graph Is this object accelerating? How do you know? What can you say about its motion?What feature of the graph represents acceleration?
47 Final velocity after any displacement (E) A baby sitter pushes a stroller from rest, accelerating at m/s2. Find the velocity after the stroller travels 4.75m. (p. 57)Identify the variables.Solve for the unknown.Substitute and solve.
49 2.3 Falling Objects 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.
51 Free FallIn the absence of air resistance, all objects fall to earth with a constant accelerationThe rate of fall is independent of massIn a vacuum, heavy objects and light objects fall at the same rate.The acceleration of a free-falling object is the acceleration of gravity, gg = 9.81m/s2 memorize this value!
52 Free FallFree 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 ag (generally) or g (on Earth’s surface).
53 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/s2.Free-fall acceleration on Earth’s surface is –9.81 m/s2 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/s2Top of path: velocity is zero, acceleration is –9.81 m/s2Moving downward: velocity is increasing, acceleration is –9.81 m/s2
54 Sample Problem Falling Object A player 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?
55 Sample Problem, continued 1. DefineGiven: Unknown:vi = +6.0 m/s Δt = ?a = –g = –9.81 m/s2Δ y = –2.0 mDiagram:Place the origin at theStarting point of the ball(yi = 0 at ti = 0).
56 2. Plan Choose an equation or situation: Both ∆t and vf are unknown. We can determine ∆t if we know vfSolve for vf then substitute & solve for ∆t3. CalculateRearrange the equation to isolate the unknowns:vf = m/sΔt = 1.50 s
57 Summary of Graphical Analysis of Linear Motion This is a graph of x vs. t for an object moving with constant velocity. The velocity is the slope of the x-t curve.
58 Comparison of v-t and x-t Curves On the left we have a graph of velocity vs. time for an object with varying velocity; on the right we have the resulting x vs. t curve. The instantaneous velocity is tangent to the curve at each point.
59 Displacement an v-t Curves The displacement, x, is the area beneath the v vs. t curve.