1. Chapter 2 Motion in One Dimension

2. Quantities in Motion Any motion involves three concepts Displacement Velocity Acceleration These concepts can be used to study objects in motion

3. Position Defined in terms of a frame of reference One dimensional, so generally the x- or y-axis Defines a starting point for the motion

4. Displacement Defined as the change in position f stands for final and i stands for initial May be represented as y if vertical Units are meters (m) in SI, centimeters (cm) in cgs or feet (ft) in US Customary

5. Displacements

6. Vector and Scalar Quantities Vector quantities need both magnitude (size) and direction to completely describe them Generally denoted by boldfaced type and an arrow over the letter + or – sign is sufficient for this chapter Scalar quantities are completely described by magnitude only

7. Displacement Isn’t Distance The displacement of an object is not the same as the distance it travels Example: Throw a ball straight up and then catch it at the same point you released it The distance is twice the height The displacement is zero

8. Speed The average speed of an object is defined as the total distance traveled divided by the total time elapsed Speed is a scalar quantity

9. Speed, cont Average speed totally ignores any variations in the object’s actual motion during the trip The total distance and the total time are all that is important SI units are m/s

10. Velocity It takes time for an object to undergo a displacement The average velocity is rate at which the displacement occurs generally use a time interval, so let t i = 0

11. Velocity continued Direction will be the same as the direction of the displacement (time interval is always positive) + or - is sufficient Units of velocity are m/s (SI), cm/s (cgs) or ft/s (US Cust.) Other units may be given in a problem, but generally will need to be converted to these

12. Speed vs. Velocity Cars on both paths have the same average velocity since they had the same displacement in the same time interval The car on the blue path will have a greater average speed since the distance it traveled is larger

13. Graphical Interpretation of Velocity Velocity can be determined from a position-time graph Average velocity equals the slope of the line joining the initial and final positions An object moving with a constant velocity will have a graph that is a straight line

14. Average Velocity, Constant The straight line indicates constant velocity The slope of the line is the value of the average velocity

15. Acceleration We have investigated various types of accelerations…. Catapult of a/c carrier Concussion Drag race rollercoaster The following slides are technical details about the concepts and equations

16. Acceleration Changing velocity means an acceleration is present Acceleration is the rate of change of the velocity Units are m/s² (SI), cm/s² (cgs), and ft/s² (US Cust)

17. Accelerations can be +, - We won’t be using the word ‘decelerating’ We will be paying close attention to the sign of the acceleration Can you determine an object’s motion from the sign of the velocity and of the acceleration?

18. Relationship Between Acceleration and Velocity constant velocity (red arrows of same size, same direction) Acceleration equals zero

19. Relationship Between Velocity and Acceleration Velocity and acceleration are in the same direction Acceleration is uniform (blue arrows maintain the same length) Velocity is increasing, speeding up (red arrows are getting longer) velocity and acceleration = same sign

20. Relationship Between Velocity and Acceleration Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the same length) Velocity is decreasing, slowing down (red arrows are getting shorter) Velocity, acceleration = opposite signs

21. Acceleration What if you flipped direction of motion in the previous slides? When speeding up (no matter what direction), the sign of the velocity and the acceleration are the same When slowing down (no matter what the direction), the sign of the velocity and the acceleration are opposite

22. Graphical Interpretation of Acceleration is the slope of the line connecting the initial and final velocities on a velocity-time graph

23. Graphical Interpretation of the Equation

24. Remember linear and quadratic equations ? linear equation y = m x + b y is variable on y-axis x is variable on x-axis M is the slope of the line B is the y intercept Quadratic equation y = a x 2 + b x + c y is variable on y-axis x is variable on x-axis a, b, c are coefficients

25. Compare to the equation for constant velocity… d = v t + d i d is position on y-axis, t is time on x-axis These are the “VARIABLES” in the equation v is slope of the line, aka velocity d i is the initial position There are 4 ‘numbers’ If you know any three, you can solve for the unknown one

26. Compare to equations for accelerated motion… d = ½ a t 2 + v i t + d i d is position on the y-axis t is time on the x-axis These are the “variables” in the equation Acceleration a, initial velocity v i and initial position d i are the ‘coefficients’ There are 6 ‘numbers’ in this equation: if you know 5, you can solve for the unknown one.

27. Summary: constant velocity Position vs. time (linear) d = v t + d i V is constant for the entire problem

28. Summary:accelerated motion Position vs time (quadratic, parabola) d = ½ a t 2 + v i t + d i Velocity vs time (linear) v = a t + v i Acceleration is whatever is given or calculated It is a specific unchanging number

29. Additional accel equations V 2 = v i 2 + 2 a (∆ d) A useful equation that doesn’t involve time

30. Problem-Solving Hints First read the entire problem Draw a diagram Choose a coordinate system, label initial and final points, indicate a positive direction for velocities and accelerations Identify all the given information Rewrite, be sure all the units are consistent Convert if necessary What are you solving for? Choose the appropriate kinematic equation

31. Problem-Solving Hints, cont Solve for the unknowns One equation, only one number will remain unknown Check your results Does the sign (direction) of the answer make sense? Does the size, magnitude of the answer make sense? Too big or small??