1 Chapter 2: Motion along a Straight Line

2 Displacement, Time, Velocity

3 One-Dimensional Motion The area of physics that we focus on is called mechanics: the study of the relationships between force, matter and motion For now we focus on kinematics: the language used to describe motion Later we will study dynamics: the relationship between motion and its causes (forces) Simplest kind of motion: 1-D motion (along a straight line) A particle is a model of moving body in absence of effects such as change of shape and rotation Velocity and acceleration are physical quantities to describe the motion of particle Velocity and acceleration are vectors

4 Position and Displacement Motion is purely translational, when there is no rotation involved. Any object that is undergoing purely translational motion can be described as a point particle (an object with no size). The position of a particle is a vector that points from the origin of a coordinate system to the location of the particle The displacement of a particle over a given time interval is a vector that points from its initial position to its final position. It is the change in position of the particle. To study the motion, we need coordinate system

5 Position and Displacement Motion of the “particle” on the dragster can be described in terms of the change in particle’s position over time interval Displacement of particle is a vector pointing from P 1 to P 2 along the x-axis

6 Average Velocity Average velocity during this time interval is a vector quantity whose x-component is the change in x divided by the time interval

7 Average Velocity Average velocity is positive when during the time interval coordinate x increased and particle moved in the positive direction If particle moves in negative x-direction during time interval, average velocity is negative

8 X-t Graph This graph is pictorial way to represent how particle position changes in time Average velocity depends only on total displacement x, not on the details of what happens during time interval t The average speed of a particle is scalar quantity that is equal to the total distance traveled divided by the total time elapsed.

9 Average Velocity

10 Instantaneous Velocity Instantaneous velocity of a particle is a vector equal to the limit of the average velocity as the time interval approaches zero. It equals the instantaneous rate of change of position with respect to time.

11 Instantaneous Velocity On a graph of position as a function of time for one- dimensional motion, the instantaneous velocity at a point is equal to the slope of the tangent to the curve at that point.

12 Instantaneous Velocity

13 Instantaneous Velocity Concept Question The graph shows position versus time for a particle undergoing 1-D motion. At which point(s) is the velocity v x positive? At which point(s) is the velocity negative? At which point(s) is the velocity zero? At which point is speed the greatest?

14 Acceleration

15 Acceleration If the velocity of an object is changing with time, then the object is undergoing an acceleration. Acceleration is a measure of the rate of change of velocity with respect to time. Acceleration is a vector quantity. In straight-line motion its only non-zero component is along the axis along which the motion takes place.

16 Average Acceleration Average Acceleration over a given time interval is defined as the change in velocity divided by the change in time. In SI units acceleration has units of m/s 2.

17 Instantaneous Acceleration  Instantaneous acceleration of an object is obtained by letting the time interval in the above definition of average acceleration become very small. Specifically, the instantaneous acceleration is the limit of the average acceleration as the time interval approaches zero:

18 Acceleration of Graphs

19 Acceleration of Graphs

20 Acceleration of Graphs

21 Constant Acceleration Motion In the special case of constant acceleration: the velocity will be a linear function of time, and the position will be a quadratic function of time. For this type of motion, the relationships between position, velocity and acceleration take on the simple forms :

22 Constant Acceleration Motion

23 Constant Acceleration Motion Position of a particle moving with constant acceleration

24 Constant Acceleration Motion Relationship between position of a particle moving with constant acceleration, and velocity and acceleration itself:

25 Freely Falling Bodies

26 Freely Falling Bodies Example of motion with constant acceleration is acceleration of a body falling under influence of the earth’s gravitation All bodies at a particular location fall with the same downward acceleration, regardless of their size and weight Idealized motion free fall: we neglect earth rotation, decrease of acceleration with decreasing altitude, air effects Galileo Galilei Aristotle B.C.E.

27 Freely Falling Bodies The constant acceleration of a freely falling body is called acceleration due to gravity, g Approximate value near earth’s surface g = 9.8 m/s 2 = 980 cm/s 2 = 32 ft/s 2 g is the magnitude of a vector, it is always positive number Exact g value varies with location Acceleration due to gravity Near the sun: 270 m/s 2 Near the moon: 1.6 m/s 2