Chapter 3

1. Chapter 3 Accelerated Motion

2. Introduction • In this chapter we will examine acceleration and define it in terms of velocity. • We will also solve problems involving constant acceleration and motion in one direction. • Finally, we will learn to interpret graphical representations of acceleration.

3. Acceleration occurs when there is a change in the _______ of movement or the ________. (Section 3.1) Introduction

4. We can solve for many problems involving motion in one direction at constant acceleration with three equations. (Section 3.2) Introduction

5. __________objects move in ____ direction and have _______ acceleration. (Section 3.3) Introduction

6. Section 3.1 Acceleration • Changing _____________ means an acceleration is present. • That means a change in ______or change in ________ will result in an acceleration. • Acceleration is the rate of change of the velocity: • The dimensions of acceleration are L/T2

7. Section 3.1 • The car undergoes acceleration because it’s velocity changes as it moves to the right. • The velocity we use to solve for acceleration is the ______________ velocity.

8. Section 3.1 Average Acceleration • _________ quantity (acceleration has direction and magnitude). • When the sign of the velocity and the acceleration are the same (either positive or negative), then the speed is ____________. • When the sign of the velocity and the acceleration are in the opposite directions, the speed is ___________.

9. Section 3.1 Sample Problem A ball starts at the bottom of a slanted driveway and is rolled up the driveway at 2.50 m/s. It rolls up the driveway for 5.00 s, stops for an instant, and then rolls back down. The positive direction is chosen to be up the driveway and the origin is at the place where the motion begins. What is the sign and magnitude of the ball’s acceleration up the driveway?

10. Section 3.1 Relationship Between Acceleration and Velocity • Uniform velocity (shown by red arrows maintaining the same size). • Acceleration equals ________.

11. Section 3.1 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 (red arrows are getting longer). • Positive velocity and positive acceleration.

12. Section 3.1 Negative Acceleration • A __________ acceleration does not necessarily mean the object is slowing down. • If the acceleration and velocity are both negative, the object is speeding up. • The sign give each is determined by the assignment of the (+) and (-) direction.

13. Section 3.1 Negative Acceleration • Acceleration and velocity are in opposite directions • Acceleration is uniform (blue arrows maintain the same length) • Velocity is decreasing (red arrows are getting shorter) • Velocity is positive and acceleration is negative

14. Section 3.1 Acceleration:Graphical Representation • Average acceleration is the ______ of the line connecting the initial and final velocities on a ____________ graph. • ________ acceleration is the slope of the ______ to the curve of the velocity-time graph.

15. Section 3.1 Constant Acceleration Graphs • The slope of a velocity-time graph gives the acceleration. • Constant acceleration = velocity changing at a constant rate.

16. Interpreting a Velocity-Time Graph (+) (+)

17. Section 3.1 Sample Problem A baseball player moves in a straight- line path in order to catch a fly ball hit to the outfield. His velocity as a function of time is shown in the graph. Find the instantaneous acceleration at points A, B, and C.

18. Section 3.1 Instantaneous and Uniform Acceleration • The limit of the average acceleration as the time interval goes to zero: • When the instantaneous accelerations are always the same, the acceleration will be __________. • The instantaneous accelerations will all be equal to the average acceleration. ∆v ∆t lim ∆t→0 a =

19. Section 3.1 Instantaneous Acceleration Graphs • This acceleration can be found by finding the slope of the line tangent to the point of time we are interested in.

20. Section 3.2 Motion in One Dimension at Constant Acceleration • __________ is the study of motion without regard for the cause of the motion. • In chapter 2 we discussed how velocity can be calculated in terms of position and time. • We can now describe the relationship between __________, _______, and __________ given three equations.

21. Section 3.2 Kinematic Equations • Used in situations with uniform acceleration and motion in one direction. vf = vi + atf ∆x = vit + 1/2atf2 vf2 = vi2 + 2a∆x

22. One-Dimensional Motion with Constant Acceleration • Each of the above equations are used to solve problems of _______________when the __________ does not change. • Many ___________ applications deal with constant acceleration. That is constant acceleration applies to many objects in the natural world. • At constant acceleration: • Instantaneous Average acceleration acceleration =

23. Section 3.2 Notes on the equations • Shows velocity as a function of acceleration and time. • Use when you don’t know and aren’t asked to find the ______________ . • “The velocity after a period of time is equal to the starting velocity plus the constant acceleration the object undergoes during that time period.” vf = vi + atf

24. Section 3.2 Sample Problem If a car accelerates from rest at a constant 5.5 m/s2, how long will it take for the car to reach a velocity of 28 m/s?

25. Section 3.2 Graphical Interpretation vf = vi + atf

26. Section 3.2 Graphical Method for Obtaining the Displacement • Displacement of an object moving at constant acceleration can be obtained from calculating the _________ under the curve of a velocity-time graph. ∆d = v x ∆t

27. Sample Problem The v-t graph above shows the motion of an airplane. Find the displacement of the airplane at ∆t = 1.0 s and at ∆t = 2.0 s.

28. Section 3.2 Graphical Method for Obtaining the Displacement • For an object undergoing a changing acceleration, use __________ and ___________ to approximate the area under the graph.

29. Section 3.2 ∆d = v x ∆t ∆d = 1/2a∆t

30. Section 3.2 Notes on the equations • Gives displacement as a function of time, velocity and acceleration. • Use when you don’t know and aren’t asked to find the _______________. ∆x = vit + 1/2atf2 xf = xi + vit + 1/2atf2

31. Section 3.2 Notes on the equations • Gives velocity as a function of acceleration and displacement. • Use when you don’t know and aren’t asked for the _________. vf2 = vi2 + 2a∆x vf2 = vi2 + 2a(xf – xi)

32. Section 3.2 Sample Problem An automobile starts from rest and speeds up at 3.5 m/s2 after the traffic light turns green. How far will it have gone when it is travelling at 25 m/s?

33. Section 3.2 Problem-Solving Hints • Read the problem. • Draw a diagram. • Choose a coordinate system, label initial and final points, indicate a positive direction for velocities and accelerations • Label all quantities, be sure all the units are consistent. • Convert if necessary • Choose the appropriate kinematic equation. • Solve for the unknowns. • You may have to solve two equations for two unknowns

34. Section 3.2 Sample Problem A race car starting from rest accelerates at a rate of 5.00 m/s2. What is the velocity of the car after it has traveled 100 ft?

35. Free Fall “The motion of a body when air resistance is negligible and the action can be considered due to _______ alone.”

36. Section 3.3 Free Falling Objects • All objects moving under the influence of gravity only are said to be in free fall. • Free fall does not depend on the object’s original motion. • All objects falling near the earth’s surface fall with a ______________________________. • The acceleration is called the acceleration due to gravity, and indicated by ___. • Free falling objects are not always ______________________________.

37. Section 3.3 Free Falling Objects • In the absence of air resistance all objects dropped near the surface of the Earth fall with the same __________________________.

38. Section 3.3 Acceleration Due to Gravity • Symbolized by g • g = ___________ • When estimating, use g» 10 m/s2 • g is always directed ___________. • Toward the center of the earth • Ignoring air resistance and assuming g doesn’t vary with altitude over short vertical distances, free fall is constantly accelerated motion. • g can __________ depending on one’s ____________ on the Earth.

39. Section 3.3 Free Falling and Problem Solving • If we neglect air resistance and assume that g does not vary, then the free fall motion approximates motion in one dimension under constant acceleration. • Up = (+) direction; Down = (-) direction • “y” replaces “x” in the kinematics equations. • g replaces “a” in these equations as well.

40. Sample Problem Suppose the ride at the right starts from rest and is in free fall for 1.5 s. What would the velocity at the end of this time? How far does the car fall?

41. Section 3.3 Three Kinds of Free Falls • We can describe three different ways that an object can undergo a free fall. • In all three cases the acceleration is the same. • What changes is __________ and the ________ traveled.

42. Section 3.3 Free Fall – An Object Dropped • Initial velocity is _______. • Let up be positive. • Use the kinematic equations. • Generally use y instead of x since vertical • Acceleration is g = -9.80 m/s2 vo= 0 a = g

43. Sample Problem A golf ball is released from rest at the top of a very tall building. Neglecting air resistance, calculate the position and the velocity of the ball after 1.00, 2.00, and 3.00 s.

44. Free Fall – An Object Thrown Downward • a = g = -9.80 m/s2 • Let up be positive • Use the kinematic equations • Generally use y instead of x since vertical • ________________ 0 • With upward being positive, initial velocity will be negative. vo= ? a = g

45. Section 3.3 Free Fall – Object Thrown Upward • Initial velocity is ________, so _________. • The instantaneous ________ at the maximum height is ________. • a = g = -9.80 m/s2 everywhere in the motion. v = 0

46. Section 3.3 Free Fall - Object Thrown Upward • The motion may be ____________. • Then tup = tdown • Then v = -vo • Actually at ____________above the origin vup = vdown . • The motion may not be symmetrical • Break the motion into various parts • Generally up and down

47. Sample Problem You decide to flip a coin to determine whether to do your physics or English homework first. The coin is flipped straight up. If the coin reaches a high point of 0.25 m above where you released it, what was its initial speed?

48. Chapter 3 Accelerated Motion The End