Welcome back to Physics 211

1. Welcome back to Physics 211 Today’s agenda: Using graphs to solve problems Constant acceleration

2. Reminder Homework due: • Wednesday: Tutorial HW on Velocity (pages 3 - 6 in HW volume)

3. Note Taker PHY211 • Office of Disability Services will pay a student for providing notes for class • $90 for semester • If you are interested please see ODS at 104 University Ave, Suite 109 x5024

4. Velocity=slope of s vs t graph s L Ds V=Ds/Dt As Dt->0 L is tangent V->vinst t Dt t

5. Displacement from velocity curve ? • Suppose I know v(t) (say as graph) can I learn anything about s(t) ? • Consider a small time interval Dt v = Ds/Dt  Ds = vDt • So total displacement is the sum of all these small displacements Ds s = SDs = S v(t)Dt =

6. Graphical interpretation v v(t) T1 T2 t Dt Displacement between T1 and T2 is Area under v(t) curve

7. Similarly for v vs t graph ainst=limDt->0 Dv/Dt=dv/dt interpreted as gradient of tangent to v vs t graph at some time If Dt held finite this definition yields the average acceleration aav over that time interval

8. Velocity from accel curve • Similarly change in velocity in some time interval is just area enclosed between curve a(t) and t axis in that interval.

9. Simplest case with acceleration • Constant accleration – a • Can find simple equations for s(t), v(t) in this case

10. V vs t graph Linear … v vI t

11. 1st constant accel. equation • From definition of aav aav=Dv/Dt Let aav=a, Dt=t, Dv=vF-vI Find: vF=vI+at equation of straight line in v vs t

12. 2nd const accel. equation • Notice: graph makes it clear that vav=1/2(vF+vI)

13. 3rd const accel. equation • Using 1st const a equation vF=vI+at insert into relation between s, t and vav: s=vavt=1/2(vF+vI)t (set Ds=s, Dt=t) Yields: s=1/2(2vI+at)t i.e s=vIt+1/2at2

14. S vs t graph s parabola Initial v>0 Sign of a ? t

15. 4th const accel. equation • Can get an equation independent of t • Substitute t=(vF-vI)/a into s=1/2(vF+vI)t we get: 2as=vF2-vI2 or vF2=vI2+2as

16. Rolling Disks demo • Compute average velocity for each section of motion (between marks) • Measure time taken • Plot V against t  straight line !

17. Motion with constant acceleration: vF= vI + a t vav= 1/2 (vI+ vF) sfinal = vIt + 1/2at2 vF2= vI2 + 2 as where “initial” refers to time = 0 s ; “final” to time t

19. Observations • Case P: • what is vav for slope section ? • what is vI at top of slope ? • what is vF at bottom of slope ? • What is accel down slope ? • what is v along flat section ?

20. An object moves with constant acceleration for 40 seconds, covering a total distance of 800 meters. Its average velocity is therefore 20 m/s. Its instantaneous velocity will be equal to 20 m/s… 1. at t = 20 seconds (i.e., half the time) 2. at s = 400 meters (i.e., half the distance) 3. both of the above. 4. Cannot tell based on the information given.

22. An object moves with constant acceleration, starting from rest at t = 0 s. In the first four seconds, it travels 10 cm. What will be the displacement of the object in the following four seconds (i.e., between t = 4 s and t = 8 s)? 1. 10 cm 2. 20 cm 3. 30 cm 4. 40 cm

23. A car is speeding up while moving in a straight line. At some point, the car has a velocity of 20 m/s. Twenty-five seconds (25 s) later, the car has moved another 600 meters and now has a velocity of 30 m/s. The average velocity of the car during this interval is: 1. 20 m/s 2. 24 m/s 3. 25 m/s 4. 30 m/s (Hint: The acceleration of the car is not constant.)

24. Vectors • are used to denote quantities that have magnitude and direction • can be added and subtracted • can be multiplied or divided by a number • can be manipulated graphically (i.e., by drawing them out) or algebraically (usually by considering components)

25. Adding vectors To add vector B to vector A: Draw vector A. Draw vector B with its tail starting from the tip of A. The sum vector A+B is the vector drawn from the tail of vector A to the tip of vector B.

27. 2D Motion S – vector position

28. Describing motion with vectors • Positions and displacements s,Ds = sF - sI • Velocities and changes in velocity: vav= ––––,vinst= lim ––––, Dv = vF - vI • Acceleration: aav= ––––,ainst= lim ––––, Dt0 Dt0