Chapter 1. Concepts of Motion Kinematics – the mathematical description and analysis of motion.

Chapter 1 Student Learning Objectives • To analyze the motion of an object by using motion diagrams as a tool. • To understand and use the basic ideas of the particle model. • To differentiate between concepts of position, velocity, and acceleration. • To recognize the relationship between and when an object is speeding up, slowing down, curving, or at a turning point. • To gain initial experience with graphical addition and subtraction of vectors. • To begin the process of learning to analyze problem statements and to translate the information into other representations. • To make and interpret graphs of motion.

Motion Diagrams – a movie without the camera A non-moving movie camera takes photographs at a fixed rate, e.g. 30 photographs (or frames) every second. Each is separated by the same time interval (here ∆t = 0.033s). frames are all lined up one after the other in a filmstrip. The moving car is in a somewhat different position in each frame.

Making a Motion Diagram Cut frames apart, stack them on top of each other. Project the entire stack at once onto a screen for viewing. This composite photo, showing an object’s position at several equally spaced instants of time, is a motion diagram.

Which car is going faster, A or B Which car is going faster, A or B? Assume there are equal intervals of time between the frames of both movies. Answer: B

Which car is going faster, A or B? Assume there are equal intervals of time between the frames of both movies. STT1.1 B is going faster.

Examples of Motion Diagrams Images equally spaced indicate constant speed Increasing distance between images mean object is speeding up

Examples of Motion Diagrams A decreasing distance between imaes shows the object is slowing down. More complex motion shows both slowing down and speeding up.

The Particle Model – A simplifying assumption Consider the moving object as if all its mass were concentrated into a single point, without size or shape. An object that can be represented as a mass at a single point in space is called a particle.

Three motion diagrams are shown Three motion diagrams are shown. Which is a dust particle settling to the floor at constant speed, which is a ball dropped from the roof of a building, and which is a descending rocket slowing to make a soft landing on Mars? A. (a) is ball, (b) is dust, (c) is rocket B. (a) is ball, (b) is rocket, (c) is dust C. (a) is rocket, (b) is dust, (c) is ball D. (a) is rocket, (b) is ball, (c) is dust E. (a) is dust, (b) is ball, (c) is rocket Answer: A

Three motion diagrams are shown Three motion diagrams are shown. Which is a dust particle settling to the floor at constant speed, which is a ball dropped from the roof of a building, and which is a descending rocket slowing to make a soft landing on Mars? A. (a) is ball, (b) is dust, (c) is rocket B. (a) is ball, (b) is rocket, (c) is dust C. (a) is rocket, (b) is dust, (c) is ball D. (a) is rocket, (b) is ball, (c) is dust E. (a) is dust, (b) is ball, (c) is rocket STT1.2

Position and time measurements for a basketball thrown in the air – using coordinates Coordinate system with origin (which is not the same as the starting point) The frame a t = 0s is frame 0, the frame at t = 0.5 s is frame 1, etc.

Position measurements – using the position vector A vector is a quantity that is described by both magnitude and direction Direction is given by the angle relative to the x axis

Tactics: Vector Addition

Tactics: Vector Subtraction

Displacement – change in position Sam is standing 50 ft east of the corner of 12th St. and Vine. He then walks northeast for 100 ft. The initial position vector r0 plus the displacement vector ∆r equals the final position vector, r1 . Note that Sam was never at the location chosen to be the origin.

Displacement – change in position ∆r = r1 - r0 Use the rules of vector subtraction (r1 + -r0) to prove that ∆r is the vector connecting the dots of a motion diagram.

Workbook Exercise #10

Average Speed, Average Velocity To quantify an object’s fastness or slowness, we define a ratio as follows: Average speed is a scalar Average velocity is a vector quantity that includes direction. The average velocity of an object during a time interval Δt, in which the object undergoes a displacement Δr, is the vector:

An object’s average velocity vector points in the same direction as the displacement vector, Δr. This is the direction of motion. Since vavg only differs from Δr by a scalar (i.e they are proportional), the vector arrows connecting each dot of a motion diagram can also be an vavg vector. We usually drop the subscript for motion diagrams.

Motion Diagrams with Velocity Vectors

Average Acceleration Because velocity is a vector, it can change in two possible ways. The magnitude can change, indicating a change in speed, or 2. The direction can change, whether or not the speed changes. Of course, both can change at the same time! We will concentrate for now on the first case, a change in speed. This is linear acceleration

Average Acceleration The average acceleration of an object during the time interval ∆t, in which the object’s velocity changes by ∆v, is the vector: aavg = ∆v/ ∆t

Average Acceleration aavg = ∆v/ ∆t The average acceleration vector is proportional to, and points in the same direction as, the change in velocity vector. This is not necessarily the direction of motion. Acceleration can refer to an object speeding up or slowing down. A “negative acceleration” means that ∆v points left, down, west or south on conventional Cartesian axes or on a map. It has nothing to do with slowing down.

Tactics: Finding the acceleration vector

Tactics: Finding the acceleration vector

A particle undergoes acceleration while moving from point 1 to point 2 A particle undergoes acceleration while moving from point 1 to point 2. Which of the choices shows the velocity vector as the object moves away from point 2?

A particle undergoes acceleration while moving from point 1 to point 2 A particle undergoes acceleration while moving from point 1 to point 2. Which of the choices shows the velocity vector as the object moves away from point 2? The answer is b

Determining the sign of position, velocity, acceleration

Velocity and Acceleration An object is speeding up if and only if v and a have the same sign. An object is slowing down if and only if v and a have opposite signs. An object is constant if and only a = 0.

A particle undergoes acceleration while moving from point 1 to point 2 A particle undergoes acceleration while moving from point 1 to point 2. Which of the choices shows the velocity vector as the object moves away from point 2? Answer: C

A particle undergoes acceleration while moving from point 1 to point 2 A particle undergoes acceleration while moving from point 1 to point 2. Which of the choices shows the velocity vector as the object moves away from point 2? STT1.4

Problem #35 in Workbook

Position vs time graphs(EOC #18) Below is a motion diagram for a drag racer Draw a position vs time graph for this motion diagram.

Answer to #18