Kinematics – Part A Physics 30S

Outcomes S3P-3-02: Differentiate among position, displacement, and distance. S3P-3-03: Differentiate between the terms “an instant” and “an interval” of time. S3P-3-04: Analyze the relationships among position, velocity, acceleration, and time for an object that is accelerating at a constant rate. Include: transformations of position-time, velocity-time, and acceleration-time graphs using slopes and areas S3P-3-05: Compare and contrast average and instantaneous velocity for non-uniform motion. Include: slopes of chords and tangents S3P-3-06: Illustrate, using velocity-time graphs of uniformly accelerated motion, that average velocity can be represented as and that displacement can be calculated as S3P-3-07: Solve problems, using combined forms of:

Instant vs. Interval of Time What is an instant in time? In physics, one point in time For example, 3:15.42.35 One glance at the clock Idealized concept An interval is a time period For example, between 10:00 am and 11:00 am 3:15.42.35 to 3:15.42.36

Measuring Time Intervals Pencil, paper, stopwatch, hard surface Ideas: helicopters, bottle pressure, trebuchet? Distance (cm) Time Average Velocity 10 20 30 40 50 60

Measuring Time Instants Pencil, paper, stopwatch, hard surface, ruler (overhead grid) Ideas: camera, rubber ball Time Position 0:00.00

What Does it Mean on a Graph? Interval: section Instant: single point More to come later...

Homework Journal entry: Compare and contrast an instant and an interval. Use your ideas to guess what the difference is between instantaneous velocity and average velocity and between instantaneous acceleration and average acceleration.

Introduction to Kinematics

The Meaning of Negatives in Kinematics Negatives indicate direction in kinematics! Standard reference system E/right is positive W/left is negative N/up is positive S/down is negative Think of a number line!

SI Units Système International des Unités (SI units) Designed for consistency between all scientists Always use SI units unless specifically instructed otherwise!!! Quantity Unit Displacement m Velocity m/s Acceleration m/s2 Time s Mass kg Force N

Position, Displacement and Distance Reminder: Position: where are you right now Distance: scalar, how far away are you from the reference point Displacement: vector, distance and a direction

Averages Averages intend to paint the “overall” picture In physics, we will discuss average velocity and average acceleration

Displacement, Velocity, and Acceleration - How Are They Related? Remember, average velocity and acceleration are taken over an interval! Δ is the Greek letter delta. It means “change in”, as in Δt is change in time.

Problem solving Example 1 What is the average acceleration of a car which speeds up to 120 km/h from 100.0 km/h in 1.5 seconds? Provide your answer in m/s2. aav = 3.7 m/s2 Example 2 What is the average velocity of a car which travels 25 m in 2.4 seconds? What is its velocity in km/h? Vav= 10. m/s

Working with Displacement, Velocity & Acceleration – Problem Solving Example 3 How far does a car travelling 100. km/h travel in 3.5 s? Provide your answer in m. d = 97 m Example 4 What is the new velocity of the car from question 3 if it accelerates at 2.2 m/s2 for 3.0 s? Vf = 34 m/s

Homework Physics, Concepts and Conceptions Pg 30 #17, 19

Displacement, Velocity and Acceleration vs. Time Graphs

Position, Velocity, Acceleration Which object is moving faster? A B

Displacement Time Graphs We can graph the displacement travelled as a function of time Use this graph to tell us about displacement, velocity and acceleration

Constant Acceleration Many objects accelerate at a constant rate Free fall (assume no air resistance) Cart being pulled by an elastic Useful to understand what constant acceleration means in terms of velocity Velocity is increasing by the same amount each time interval 5 km/h, 10 km/h, 15 km/h, 20 km/h, etc.

Constant Acceleration

Slope Slope is rise over run, the change in y over x In a displacement time graph, y is displacement and x is time Remember velocity = displacement/time In a displacement time graph, the slope is velocity In a velocity time graph, the slope is acceleration

Constant Acceleration Slope Constant Acceleration Slope

Area The area under a graph is A = lw A = xy For a velocity time graph, y is velocity, x is time A = vt But d = vt, so The area under a velocity time graph is the displacement!

In general... The formula for the area under a graph will change depending on the shape of the graph We will always work with either constant velocity (square vt graph) or constant acceleration (triangle vt graph).

Constant Acceleration Slope Area Constant Acceleration Slope Area

Graph Questions – Example 1 Imagine the d t graph provides the displacement for a rollercoaster. To two sig figs, what is the velocity of the rollercoaster? V = 1.5 m/s

Graph Questions – Example 2 Imagine the v t graph provides the velocity for a new rollercoaster dropping down the track. To three sig figs, what is the acceleration of the rollercoaster? a = 1.05 m/s2 Time (s) Velocity (m/s) 1.00 1.05 2.00 2.10 3.00 3.15 4.00 4.20 5.00 5.25 6.00 6.30 7.00 7.35 8.00 8.40

Graph Questions – Example 3 Imagine the v t graph provides the velocity for a rollercoaster dropping down the track. What is the displacement of the rollercoaster at 3.00s? d = 4.73m Time (s) Velocity (m/s) 1.00 1.05 2.00 2.10 3.00 3.15 4.00 4.20 5.00 5.25 6.00 6.30 7.00 7.35 8.00 8.40

Graph Questions – Example 4 Imagine the a t graph provides the acceleration for a rollercoaster dropping down the track. What is the velocity of the rollercoaster at 6.00s? v = 13.2m/s

Graph Questions – Example 5 Time (s) Displacement (m) 0.00 1.00 1.50 2.00 3.00 4.50 4.00 2.25 5.00 6.00 -2.25 7.00 -4.50 8.00 -6.75 9.00 10.00 11.00 -3.38 12.00 Imagine the d t graph provides the displacement for the rollercoaster’s track. a)Explain what the rollercoaster is doing through each interval. b) What is the velocity during each interval?

Graph Questions – Example 5 Solution a)The rollercoaster travels at 1.50 m/s for 3.0 s, changes direction and heads back towards the start, travelling at -2.25 m/s until 8.0s. The rollercoaster then stops until 9.0s, and then returns to the start at 3.38 m/s. b) V1 = 1.50 m/s, V2 = -2.25 m/s, V3 = 0.00 m/s, V4 = 3.38 m/s

Homework Physics, Concepts and Conceptions P.32 #28, 29, 30, 31

What about Instantaneous? Imagine taking a smaller and smaller interval: 1 s, 0.1 s, 0.01s, 0.001s, 0.0001s, etc Repeating this infinitely, there would be no interval at all – you would have an instant The intersection points would merge together to create... A point of tangency!!! Instantaneous velocity.ggb file

Tangent to the Curve The slope of the tangent line to the displacement graph at a given point in time is the instantaneous velocity at that given point in time Same relationship exists between velocity and instantaneous acceleration

Example 1 Find the instantaneous acceleration at 2.00s.

Example 1 Find the instantaneous acceleration at 4.00s. Pt 1(2.00,0.00) Pt 2(6.50, 36.00) a = 8.00 m/s2

Preview to Calculus How do you tell when you’ve got the correct tangent line? Tangent lines are difficult to draw – do your best Tangent lines are like an estimation To find the true instantaneous velocity or acceleration, we need calculus!

Homework Physics, Concepts and Conceptions P.34 # 36, 37, 38, 39, 43

Bringing Formulas into it S3P-3-06: Illustrate, using velocity-time graphs of uniformly accelerated motion, that average velocity can be represented as and that displacement can be calculated as Think back to the graph! The first formula is just slope! The second is just the first formula where we are solving for d!

Problem Solving Here are the formulas we have available: Pick one based on what you want to know and what is already known! (We are assuming constant acceleration for #2) S3P-3-07: Solve problems, using combined forms of:

Example 1 A puppy starts off at rest and accelerates at a constant rate to 8.0 m/s. Find his average velocity. 4.0 m/s

Example 2 A puppy starts off at rest and accelerates at a constant rate. If his displacement is 250 m north after 40 s, find his average velocity. 6.25 m/s

Example 3 A BMW stopped at a red light accelerates up to 110.0 km in 4.800 s. What is its average acceleration? 6.36 m/s2

Homework Basic Kinematics worksheet

Lab

The Plan Max Classes: 5 Time Intervals and Instants; Intro to kinematics (displacement, velocity, acceleration) dt, vt, at graphs Instantaneous acceleration Formulas and Problem solving Quiz