1. Motion in One Dimension
Chapter 2
Motion in One Dimension

2. Dynamics
The branch of physics involving the motion of an object and the relationship between that motion and other physics concepts
Kinematics is a part of dynamics
In kinematics, you are interested in the description of motion
Not concerned with the cause of the motion

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

4. 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

5. 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

6. Displacements

7. 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

8. 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

9. 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

10. 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

11. 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 ti = 0

12. 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

13. 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

14. 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

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

16. Average Velocity, Non Constant
The motion is non-constant velocity
The average velocity is the slope of the blue line joining two points

17. Instantaneous Velocity
The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero
The instantaneous velocity indicates what is happening at every point of time

18. Instantaneous Velocity on a Graph
The slope of the line tangent to the position-vs.-time graph is defined to be the instantaneous velocity at that time
The instantaneous speed is defined as the magnitude of the instantaneous velocity

19. Uniform Velocity Uniform velocity is constant velocity
The instantaneous velocities are always the same
All the instantaneous velocities will also equal the average velocity

20. Acceleration
Changing velocity (non-uniform) 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)

21. Average Acceleration Vector quantity
When the sign of the velocity and the acceleration are the same (either positive or negative), then the speed is increasing
When the sign of the velocity and the acceleration are in the opposite directions, the speed is decreasing

22. 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 uniform
The instantaneous accelerations will all be equal to the average acceleration

23. Graphical Interpretation of Acceleration
Average acceleration is the slope of the line connecting the initial and final velocities on a velocity-time graph
Instantaneous acceleration is the slope of the tangent to the curve of the velocity-time graph

24. Average Acceleration

25. Relationship Between Acceleration and Velocity
Uniform velocity (shown by red arrows maintaining the same size)
Acceleration equals zero

26. 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

27. Relationship Between Velocity and 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

28. Kinematic Equations
Used in situations with uniform acceleration

29. Notes on the equations
Gives displacement as a function of velocity and time
Use when you don’t know and aren’t asked for the acceleration

30. 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 displacement

31. Graphical Interpretation of the Equation

32. 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 final velocity

33. 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 time

34. 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

35. Problem-Solving Hints, cont
Solve for the unknowns
You may have to solve two equations for two unknowns
Check your results
Estimate and compare
Check units

36. Example 1
A bicyclist rides her bike at a speed of 10 Km/hr for 25 min, remembers that she forgot something at home she turns around and pedals at a rate of 12Km/hr back to her house. If turning around time is ignored what are the following?
The distance she was away from home when she decided to turn around.
The time that it took her to return home after she remembered that she forgot something.
The average speed of the entire trip.
The average velocity of the entire trip.

37. Example 2
A motorist drives south for 45.0 minutes at 75.0 km/h and then stops for 20.0 minutes. He then continues south, traveling 120 km in 2.50 h. (a) What is his total displacement? (b) What is his average velocity?

38. Example 3
A certain horse ran a race with the following times: 1st quarter 24.5s, 2nd quarter 23.5s, 3rd quarter 22.4s and the 4th quarter time was 24.2s. (a) Find the average speed during each quarter-mile segment. (b) Assuming that the horse’s instantaneous speed at the finish line was the same as the average speed during the final quarter mile, find the average acceleration for the entire race. (Hint: Recall that horses in the race start from rest.)

39. Example 4
A certain car is capable of accelerating at a rate of m/s2. How long does it take for this car to go from a speed of 45 mi/h to a speed of 60 mi/h?

40. Example 5
Two cars are traveling along a straight line in the same direction, the lead car at 35.0 m/s and the other car at 45.0 m/s. At the moment the cars are 50.0 m apart, the lead driver applies the brakes, causing his car to have an acceleration of –1.50 m/s2. (a) How long does it take for the lead car to stop? (b) Assuming that the chasing car brakes at the same time as the lead car, what must be the chasing car’s minimum negative acceleration so as not to hit the lead car? (c) How long does it take for the chasing car to stop?

41. Example 6
A drag racer starts her car from rest and accelerates at m/s2 for a distance of 440 m. (a) How long did it take the race car to travel this distance? (b) What is the speed of the race car at the end of the run?

42. Example 7
A car traveling in a straight line has a velocity of +5.0 m/s at some instant. After 4.0 s, its velocity is +8.0 m/s. What is the car’s average acceleration during the 4.0-s time interval?

43. Example 8
a) What is the average velocity of the object during the 0 to 4s time interval?
b) What is the instantaneous velocity of the object at point A?
c) What is the average velocity of the object during the 8 to 12s time interval?

44. Galileo Galilei
Galileo formulated the laws that govern the motion of objects in free fall
Also looked at:
Inclined planes
Relative motion
Thermometers
Pendulum

45. Free Fall
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 constant acceleration
The acceleration is called the acceleration due to gravity, and indicated by g

46. Elephant and Feather

47. Acceleration due to Gravity
Symbolized by g
g = m/s²
When estimating, use g » -10 m/s2
g is always directed downward
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

48. Free Fall – an object dropped
Initial velocity is zero
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

49. Example 9
A rock is dropped from the top of a cliff and lands at the base of the cliff 3.2s after it is released. How far did the rock drop?

50. Example 10
Another scheme to catch the roadrunner has failed! Now a safe falls from rest from the top of a 25.0-m-high cliff toward Wile E. Coyote, who is standing at the base. Wile first notices the safe after it has fallen 15.0 m. How long does he have to get out of the way?

51. Example 11
A stunt man sitting on a tree limb wishes to drop vertically onto a horse galloping under the tree. The constant speed of the horse is 10.0 m/s, and the man is initially 3.00 m above the level of the saddle. (a) What must be the horizontal distance between the saddle and the limb when the man makes his move? (b) How long is he in the air?

52. Free Fall – an object thrown downward
a = g = m/s2
Initial velocity  0
With upward being positive, initial velocity will be negative

53. Example 13
A student standing on the top of the science building on campus throws a water balloon at his physics instructor. If the initial velocity of the balloon is -2.3m/s, and it just misses the instructor and lands on the ground in front of him 1.2s after it was released from what height was the balloon released? How fast was it moving when it struck the ground?

54. Free Fall -- object thrown upward
Initial velocity is upward, so positive
The instantaneous velocity at the maximum height is zero
a = g = m/s2 everywhere in the motion
v = 0

55. Thrown upward, cont. The motion may be symmetrical
Then tup = tdown
Then v = -vo
The motion may not be symmetrical
Break the motion into various parts
Generally up and down

56. Example 14
It is possible to shoot an arrow at a speed as high as 100 m/s. (a) If friction is neglected, how high would an arrow launched at this speed rise if shot straight up? (b) How long would the arrow be in the air?

57. Example 15
A ball is thrown vertically upward with a speed of 35.0 m/s. (a) Where is the ball 4.00 seconds after it was released? (b) What is its velocity at the time? (c) Has it reached its maximum height yet? Defend your answer.

58. Non-symmetrical Free Fall
Need to divide the motion into segments
Possibilities include
Upward and downward portions
The symmetrical portion back to the release point and then the non-symmetrical portion

59. Example 16
A ball is thrown upward with a velocity of +4m/s from the edge of a 10m building. a) How long does it take the ball to reach its maximum height? b) What is the maximum height of the ball? c) If the ball just misses the edge of the building on the way back down how long will the ball be in the air? d) How fast is the ball moving when it strikes the ground?

60. Combination Motions

61. Example 17
A model rocket is launched straight upward with an initial speed of 50.0 m/s. It accelerates with a constant upward acceleration of 2.00 m/s2 until its engines stop at an altitude of 150 m. (a) What is the maximum height reached by the rocket? (b) How long after lift-off does the rocket reach its maximum height? (c) How long is the rocket in the air?