Kinematics in One Dimension

1. Kinematics in One Dimension Position Vector Displacement Vector, Distance Velocity Vector Acceleration Vector

2. Kinematics • Geometrical and algebraic description of motion • No regard to the causes of motion (forces) • Makes use of the mathematical concept of coordinate system attached to a point in space or to objects called ‘frames of reference’. • Coordinate system is used to describe position relative to the origin. • Location of origin is arbitrarily selected.

3. +1.5 m x (m) -2 -1 O +1 +2 One Dimensional Coordinate System • When dealing with a single dimension the coordinate system is a straight line. • Points on the line are located relative to an arbitrary point called the origin. • Distance on one side of the origin is positive; negative on the opposite side. • A point in 1D space corresponds to a point on the line; a positive or negative number with units of length. Positive and negative axis directions are arbitrarily selected; usually the positive direction is chosen to the right. Thus a point can be described by a single signed coordinate value with unit. • Sample point is 1.5 m on the positive side of the origin.

4. +1.5 m x (m) -2 -1 O +1 +2 Position as a Vector • A point can also be located by a position vector. • Position vector has tail at the origin and tip at the point. • The length of the vector is the distance (coordinate value) of the point from the origin. • The vector direction corresponds to the sign of the coordinate value.

5. Displacement has unit of length such as m. Displacement Vector - describes the change in position of a moving object • Vector pointing from an object’s initial position to its final position. In one dimension position and displacement, although vectors, can be unambiguously described by scalars.

6. Positive and Negative Displacement

7. Distance • Magnitude of displacement if motion is in one direction only (no reversal of direction). • Distance is always positive because it is has magnitude only with no direction. • If motion consists of a sequence of positive and negative displacements, the distance is the sum of distances for each of the segments of the displacement. 4.0 m 4.0 m 2.0 m x = + 4.0 m d = 4.0 m x = + 2.0 m d = 4.0m + 2.0 m = 6.0 m

8. Distance versus Displacement • What if you travel around the running track? • What is your displacement? • What is your distance?

9. Speed has units of length per unit time such as m/s. Average Speed - a scalar quantity • A stage of the Tour de France from Melun to Paris has a distance of 140 km. • Australian, Robbie McEwen, won the stage by cycling the distance in 3h, 30 min and 47 s. What was his average speed?

10. Velocity has units of length per unit time such as m/s. Average Velocity - a vector quantity A man taking a leisurely walk takes 15 minutes to walk 100 m in an eastward direction. He then walks 50 m back (west) in 5 minutes. What is his average speed and velocity in m/s? 100 m E W 50 m

11. Instantaneous Velocity - average velocity when the elapsed time approaches zero Instantaneous Velocity or simply velocity Instantaneous Speed or simply speed - the magnitude of the velocity. If v is constant, the x-t graph is that of a straight line. The average and instantaneous velocities are the same and is equal to the slope of the line. If v is not constant the instantaneous velocity at a time t is the slope of the curved x-t graph. Instantaneous and average velocities are not necessarily the same (although they can be).

12. Small intervals give you instants

13. Meaning of the sign of velocity Positive velocity means direction of v is towards positive direction of x axis, i.e., displacement is positive. Slope of x-t graph will be positive. Negative velocity means direction of v is towards negative x axis, i.e., displacement is negative. Slope of x-t graph will be negative. The sign of v does not indicate whether speed is increasing or decreasing. It, however, indicates what is happening to the displacement.

14. Meaning of the sign of velocity • You can consider the world to have a positive direction and a negative direction. • Are you going towards POSIWORLD? • Or NEGILAND?

15. Acceleration - a vector quantity Average Acceleration Unit: length/time2 such as m/s2 Instantaneous Acceleration - average acceleration when the elapsed time approaches zero.

16. Acceleration and Velocity Whether an object is speeding up or slowing down does not depend on the sign of a but on its direction relative to the direction of v.

17. a and v in the same direction

18. a and v in opposite directions Dt = 3 s

19. a and v in opposite directions Dt = 3 s When acceleration and velocity ‘compete’ then the object is slowing down

20. Kinematic Equations for Constant Acceleration We already have one kinematic equation (previous slide) The graph of v vs. t if a is constant is a straight line. v vavg v0 slope = a t Usually, we assume initial time t0 = 0 and initial position x0 = 0.

21. Kinematic Equations . . .

22. Kinematics Equations

23. Summary of Kinematic Equations in 1 D constant a Remember, kinematic variables are vector quantities. Their signs are important. Also x is a displacement from the initial position which was assumed to be zero. Initial time was also assumed zero.

24. Freely Falling Bodies Example of motion with constant acceleration. Motion is vertical with acceleration of gravity, g = 9.81 m/s2 always downward. Coordinate axis will be vertical (you can call it x or y). Choose positive direction up or down but be sure the sign of a is correct. If you choose upward is positive, then a = -g. If you choose downward as positive, then a = +g.

25. Be sure to get study packet and we will work some problems out in class.