Chapter 2: One-Dimensional Motion

1. Chapter 2: One-Dimensional Motion • Motion at fixed velocity • Definition of average velocity • Motion with fixed acceleration • Graphical representations

2. Displacement vs. position Position: x (relative to origin) Displacement: Dx = xf-xi

3. Average velocity Average velocity • Can be positive or negative • Depends only on initial/final positions • E.g., if you return to original position, average velocity is zero

4. Instantaneous velocity Let time interval approach zero • Defined for every instance in time • Equals average velocity if v = constant • SPEED is absolute value of velocity

5. Graphical Representation of Average Velocity Between A and D , v is slope of blue line

6. Graphical Representation of Instantaneous Velocity v(t=3.0) is slope of tangent (green line)

7. Example 2.1 Carol starts at a position x(t=0) = 1.5 m. At t=2.0 s, Carol’s position is x(t=2 s)=4.5 m At t=4.0 s, Carol’s position is x(t=4 s)=-2.5 m • What is Carol’s average velocity between t=0 and t=2 s? • What is Carol’s average velocity between t=2 and t=4 s? • What is Carol’s average velocity between t=0 and t=4 s? a) 1.5 m/s b) -3.5 m/s c) -1.0 m/s

8. Example 2.2 On a mission to rid Spartan Stadium of vermin, an archer shoots an arrow across the stadium at an unlucky rat 200 meters away. The archer hears the squeal 2.2 seconds later. What was the velocity of the arrow? The speed of sound is 330 m/s.

9. Example 2.2: Visualize the problem!

10. Example 2.2 On a mission to rid Spartan Stadium of vermin, an archer shoots an arrow across the stadium at an unlucky rat 200 meters away. The archer hears the squeal 2.2 seconds later. What was the velocity of the arrow? The speed of sound is 330 m/s. V = 125 m/s

11. Example 2.3 • The instantaneous velocityis zero at zero at a? b? c? d? e? • The instantaneous velocity is negative at a? b? c? d? e? • The average velocity is zero in the interval(s)a-b? a-c? b-e? c-e? d-e? • The average velocity is negative in the interval(s)a-b? a-c? b-e? c-e? d-e? A) b,dB) cC) c-eD) a-c,b-e

12. SPEED • speed is |v| and is always positive • average speed is sum over |Dx| elements divided by t

13. 8 D 6 4 E B 2 C A 0 0 2 4 6 8 10 12 Example 2.3a x (m) a) What is the average velocity between B and E? b) What is the average speed between B and E? t (s) a) 0.2 m/s b) 1.2 m/s

14. Acceleration The rate of change of the velocity Average acceleration: measured over finite time interval Instantaneous acceleration: measured over infinitesimal interval, Dt -> 0

15. Graphical Description of Acceleration Acceleration is slope tangent line in v vs. t graph

16. a < 0 a > 0 Graphical Description of Acceleration a is positive/negativewhen v vs. t is rising/fallingor when x vs t curves upwards/downwards

17. e b c d a Example 2.4 At which point(s) does the position equal zero? At which point(s) does the velocity equal zero? At which point(s) does the acceleration equal zero? In which segment(s) is the velocity negative? In which segment(s) is the acceleration negative? a,db,dcb-c,c-da-b,b-c

18. Constant Acceleration • a vs. t is a constant • v vs t is a straight line • x vs t is a parabola Eq.s of Motion

19. Constant Acceleration One more useful equationEliminate t by inserting Eq. (1)into Eq. (2)

20. Final List of 1-d Equations Which one should I use?Each Eq. has 4 of the 5 variables:Dx, t, v0, v & a Ask yourself “Which variable am I not given and not interested in?” If that variable is t, use Eq. (5).

21. Example 2.5 A drag racer starts her car from rest and accelerates at 10.0 m/s2 for the entire distance of a 400 m (1/4 mi) race. a) How long did it take her to finish the race? b) What is her final speed? a) 8.94 s b) 89.4 m/s

22. Free Fall • Objects under the influence of gravity (no resistance) fall with constant downward acceleration (if near Earth’s surface). g = 9.81 m/s2 • Use the usual equations with a --> -g

23. Galileo • Father was a musician, experimented with music • Initially was a professor teaching pre-meds • Developed telescope ~ 1610: Milky Way = stars Moons of Jupiter Phases of Venus… • Measured g • Quantified mechanics • In 1632, published Dialogue concerning the two greatest world systems • Was found guilty of heresy

24. A B c Example 2.6a A man drops a brick off the top of a 50-m building. The brick has zero initial velocity. a) How much time is required for the brick to hit the ground?b) What is the velocity of thebrick when it hits the ground? a) 3.19 s b) -31.3 m/s

25. A B c Example 2.6b A man throws a brick upward from the top of a 50 m building. The brick has an initial upward velocity of 20 m/s. a) How high above the building does the brick get before it falls? b) How much time does the brick spend going upwards? c) What is the velocity of the brick when it passes the man going downwards? d) What is the velocity of the brick when it hits the ground? e) At what time does the brick hit the ground?

26. Example 2.6b A man throws a brick upward from the top of a 50 m building. The brick has an initial upward velocity of 20 m/s. a) How high above the building does the brick get before it falls? b) How much time does the brick spend going upwards? c) What is the velocity of the brick when it passes the man going downwards? d) What is the velocity of the brick when it hits the ground? e) At what time does the brick hit the ground? a) 20.4 m b) 2.04 s c) -20 m/s d) -37.2 m/s e) 5.83 s

27. B Example 2.7 A man throws a brick upward from the top of a building. TRUE OR FALSE. (Assume the coordinate system is defined with positve being upward) a) At ‘A’ the acceleration is positive b) At ‘B’ the velocity is zero c) At ‘B’ the acceleration is zero d) At ‘C’ the velocity is negative e) At ‘C’ the acceleration is negative f) The speed at ‘C’ and at ‘A’ are equal g) The velocity at ‘C’ and at ‘A’ are equal h) The speed is greatest at ‘E’ A C C A D D E

28. Example 2.7 A man throws a brick upward from the top of a building. TRUE OR FALSE. (Assume the coordinate system is defined with positve being upward) a) At ‘A’ the acceleration is positive b) At ‘B’ the velocity is zero c) At ‘B’ the acceleration is zero d) At ‘C’ the velocity is negative e) At ‘C’ the acceleration is negative f) The speed at ‘C’ and at ‘A’ are equal g) The velocity at ‘C’ and at ‘A’ are equal h) The speed is greatest at ‘E’ F T F T T T F T