1. Monday, February 7 PHYS161 Chapter 2

2. Quiz #2 TOPICS Uncertainties Uncertainties Identifying appropriate uncertainty values Identifying appropriate uncertainty values Carrying uncertainties through calculations Carrying uncertainties through calculations Motion diagrams Motion diagrams Converting verbal descriptions to motion diagrams and visa- versa Converting verbal descriptions to motion diagrams and visa- versa Identifying directions of velocity and acceleration for various situations Identifying directions of velocity and acceleration for various situations Given two vectors, determine difference and sum Given two vectors, determine difference and sum Meaning of “constant” acceleration Meaning of “constant” acceleration Graphs Graphs position vs. velocity vs. acceleration position vs. velocity vs. acceleration Equations for slope vs. definitions of velocity and acceleration Equations for slope vs. definitions of velocity and acceleration

3. Uncertainties Identifying appropriate uncertainty values Identifying appropriate uncertainty values Resolution (rounding) Resolution (rounding) Precision (repeatability) Precision (repeatability) Carrying uncertainties through calculations Carrying uncertainties through calculations For addition and subtraction, just add numerical uncertainties For addition and subtraction, just add numerical uncertainties For multiplication or division, add relative uncertainties For multiplication or division, add relative uncertainties convert numerical uncertainties to relative uncertainties convert numerical uncertainties to relative uncertainties add relative uncertainties add relative uncertainties convert total back to numerical uncertainty convert total back to numerical uncertainty

4. Examples What is the likely uncertainty associated with the resolution of the following measurements? What is the likely uncertainty associated with the resolution of the following measurements? A device provides you with a measurement value of 2.54 g. A device provides you with a measurement value of 2.54 g. A ruler with marks every millimeter. A ruler with marks every millimeter.

5. Examples You have three measurements: a = (1.0 m  0.01 m) b = (2.0 m  0.01 m) c = (2.0 m  0.02 m) What is a+b-c? What is abc? What is b 2 /c? What is √(bc)?

6. Motion diagrams Location at regular time intervals is represented by dots. The average velocity during each time interval is represented by an arrow between the two dots The average acceleration for the double time interval centered on each dot is represented by an arrow coincident with each dot

7. Directions The direction of the velocity is in the direction of motion. The direction of the acceleration is in the direction of the change in velocity. Acceleration is same direction as velocity if object is speeding up Acceleration is opposite direction of velocity if object is slowing down Acceleration is perpendicular to direction of velocity if object is changing directions (and maintaining the same speed)

8. Examples A rock is thrown up in the air. On the way up, it slows down. What is the direction of the rock’s acceleration as it slows down? When the cart rolls down the incline, is the acceleration of the cart constant? Is the velocity of the cart constant? Is it possible for an object’s acceleration to be constant in direction and magnitude yet the velocity decrease? Is it possible for an object’s acceleration to be constant in direction and decreasing in magnitude yet the velocity increase?

9. Definitions Velocity = the rate at which the position changes v = ds/dt Average velocity = the average rate at which the position changes during a time interval v avg = Δs/Δt

10. Definitions Acceleration = the rate at which the velocity changes a = dv/dt Average acceleration = the average rate at which the velocity changes during a time interval a avg = Δv/Δt

11. Examples At a particular time, a car is traveling north at a speed of 20 mph. An hour later, the car is 10 miles east of the starting point, traveling south at a speed of 20 mph. 1. What is the instantaneous velocity of the car at the initial time? 2. What is the average velocity of the car during the hour? 3. What is the average acceleration of the car during the hour?

12. Example Drop a ball such that the time is recorded in three places (t 1, t 2 and t 3 ): 1. What is the average velocity of the ball during the time interval from t 1 to t 2 ? 2. What is the average velocity of the ball during the time interval from t 1 to t 3 ? 3. Assuming the acceleration is constant, at what time does the ball have an instantaneous velocity equal to the value obtained in #1? How about an instantaneous velocity equal to the value obtained in #2? 4. Assuming the acceleration is constant, what is the acceleration of the ball as it fell?

13. Graphs Given a graph of a straight line, determine the equation that represents the relationship between the two variables Given a graph of a straight line, determine the equation that represents the relationship between the two variables Given position vs. time graph, determine velocity graph. Similarly, given velocity vs. time graph, determine acceleration graph. Given position vs. time graph, determine velocity graph. Similarly, given velocity vs. time graph, determine acceleration graph. Determine what shape of a position vs. time graph is consistent with a particular velocity vs. time graph. Similarly, given acceleration vs. time graph, determine shape of velocity graph. Determine what shape of a position vs. time graph is consistent with a particular velocity vs. time graph. Similarly, given acceleration vs. time graph, determine shape of velocity graph.

14. Slope Slope (from one point to the another point) = change in vertical value divided by change in horizontal value Using y for the vertical value and x for the horizontal value, slope (from one point to the another point) = Δy/Δx

15. Slope for position vs. time graphs Using y for the vertical value and x for the horizontal value, slope (from one point to the another point) = Δy/Δx If we plot position (s) as the vertical value and time (t) as the horizontal value then slope (during the time interval Δt) = Δs/Δt

16. Slope and velocity If we plot position (s) as the vertical value and time (t) as the horizontal value then slope (during the time interval Δt) = Δs/Δt If we define average velocity as the rate at which the position changes during a time interval (Δs/Δt) then the slope of a position vs. time graph (during the time interval Δt) = v avg (during the time interval)

17. Slope at a point Slope (at a single point) = infinitesimal change in vertical value at that point divided by infinitesimal change in horizontal value at that point = dy/dx For position vs. time graph, the slope (at an instant) = ds/dt = v (at that instant)