1. Chapter Four: Motion 4.1 Position, Speed and Velocity
4.2 Graphs of Motion
4.3 Acceleration

2. 4.1 Position, Speed and Velocity
Position is a variable given relative to an origin.
The origin is the place where position equals 0.
The position of this car at 50 cm describes where the car is relative to the track.

3. 4.1 Position, Speed and Velocity
Position and distance are similar but not the same.
If the car moves a distance of 20 cm to the right, its new position will be 70 cm from its origin.
Distance = 20 cm
New position

4. 4.1 Position, Speed and Velocity
The variable speed describes how quickly something moves.
To calculate the speed of a moving object divide the distance it moves by the time it takes to move.

6. 4.1 Position, Speed and Velocity
The units for speed are distance units over time units.
This table shows different units commonly used for speed.

7. 4.1 Average speed
When you divide the total distance of a trip by the time taken you get the average speed.
On this driving trip around Chicago, the car traveled and average of 100 km/h.

8. 4.1 Instantaneous speed
A speedometer shows a car’s instantaneous speed.
The instantaneous speed is the actual speed an object has at any moment.

9. Solving Problems
How far do you go if you drive for two hours at a speed of 100 km/h?
Looking for:
…distance
Given:
…speed = 100 km/h time = 2 h
Relationships:
d = vt
Solution:
d = 100 km/h x 2 h = 200 km
= 200 km

10. 4.1 Vectors and velocity Position uses positive and negative numbers.
Positive numbers are for positions to the right of the origin and negative numbers are for positions to the left the origin.

11. 4.1 Vectors and velocity
Distance is either zero or a positive value.

12. 4.1 Vectors and velocity
We use the term velocity to mean speed with direction.

14. 4.1 Keeping track of where you are
Pathfinder is a small robot sent to explore Mars.
It landed on Mars in 1997.
Where is Pathfinder now?

15. 4.1 Keeping track of where you are
Pathfinder keeps track of its velocity vector and uses a clock.
Suppose Pathfinder moves forward at 0.2 m/s for 10 seconds.
What is Pathfinder’s velocity?

16. 4.1 Keeping track of where you are
Suppose Pathfinder goes backward at 0.2 m/s for 4 seconds.
What is Pathfinder’s change in position?

17. 4.1 Keeping track of where you are
The change in position is the velocity multiplied by the time.

18. 4.1 Keeping track of where you are
Each change in position is added up using positive and negative numbers.
Pathfinder has a computer to do this.

19. 4.1 Maps and coordinates
If Pathfinder was crawling on a straight board, it would have only two choices for direction.
Out on the surface of Mars, Pathfinder has more choices.
The possible directions include north, east, south, and west, and anything in between.

20. 4.1 Maps and coordinates
A graph using north−south and east−west axes can accurately show where Pathfinder is.
This kind of graph is called a map.
Street maps often use letters and numbers for coordinates.

21. 4.1 Vectors on a map Where are you compared to where you started?
Suppose you run east for 10 seconds at a speed of 2 m/s.
Then you turn and run south at the same speed for 10 more seconds.
Where are you compared to where you started?

22. 4.1 Vectors on a map
To get the answer, you figure out your east−west changes and your north−south changes separately.
origin = (0 , 0)

23. 4.1 Vectors on a map
Your first movement has a velocity vector of +2 m/s, west-east (x-axis).
After 10 seconds your change in position is +20 meters (east on x-axis).
d = v x t d = 2 m/s x 10 s = +20 m

24. 4.1 Vectors on a map
Your second movement has a velocity vector of −2 m/s north−south (y-axis)
In 10 seconds you move −20 meters (south is negative on y-axis)
d = 2 m/s x 10 s = -20 m
New position = (+20 , -20)

25. Solving Problems
A train travels at 100 km/h heading east to reach a town in 4 hours. The train then reverses and heads west at 50 km/h for 4 hours. What is the train’s position now?
Looking for:
…train’s new position
Given:
…velocity = km/h, east ; time = 4 h
…velocity = km/h, west ; time = 4 h
Relationships:
change in position = velocity × time

26. Solution: Solving Problems 1st change in position:
(+100 km/h) × (4 h) = +400 km
2nd change in position:
(−50 km/h) × (4 h) = −200 km
Final position:
(+400 km) + (−200 km) = +200 km
The train is 200 km east of where it started.

27. 4.2 Graphs of Motion Constant speed means the speed stays the same.
An object moving at a constant speed always creates a position vs. time graph that is a straight line.

28. 4.2 Graphs of Motion
The data shows the runner took 10 seconds to run each 50-meter segment.
Because the time was the same for each segment, you know the speed was the same for each segment.

32. 4.2 Graphs of Motion
You can use position vs. time graphs to compare the motion of different objects.
The steeper line on a position vs. time graph means a faster speed.

33. 4.2 Slope The steepness of a line is measured by finding its slope.
The slope of a line is the ratio of the “rise” to the “run”.

34. 4.2 Graphs of changing motion
Objects rarely move at the same speed for a long period of time.
A speed vs. time graph is also useful for showing the motion of an object that is speeding up or slowing down.

35. 4.2 Graphs of changing motion
On the graph, the length is equal to the time and the height is equal to the speed.
The area of the rectangle is equal to its length times its height.
Suppose we draw a rectangle on the speed vs. time graph between the x-axis and the line showing the speed.

38. 4.3 Acceleration
Acceleration is the rate at which your speed (or velocity) changes.

39. 4.3 Acceleration
What is the bike’s acceleration?

40. 4.3 Acceleration Acceleration describes how quickly speed changes.
Acceleration is the change in speed divided by the change in time.

41. 4.3 Speed and acceleration
An acceleration of 20 km/h/s means that the speed increases by 20 km/h each second.
The units for time in acceleration are often expressed as “seconds squared” and written as s2.
Can you convert this rate using conversion factors?

42. A strong wind increases its speed to 4 m/s in 3 s.
Solving Problems
A sailboat moves at 1 m/s.
A strong wind increases its speed to 4 m/s in 3 s.
Calculate acceleration.

43. = 1 m/s2 Solving Problems Looking for: …acceleration of sailboat
Given:
…v1 = 1 m/s; v2 = 4 m/s; time = 3 s
Relationships:
a = v2 – v1/t
Solution:
a = (4 m/s – 1 m/s)/ 3 s
= 1 m/s2

44. 4.3 Acceleration on speed-time graphs
Positive acceleration adds more speed each second.
Things get faster.
Speed increases over time.

45. 4.3 Acceleration on speed-time graphs
Negative acceleration subtracts some speed each second.
Things get slower.
People sometimes use the word deceleration to describe slowing down.

46. 4.3 Acceleration on position-time graphs
The position vs. time graph is a curve when there is acceleration.
The car covers more distance each second, so the position vs. time graph gets steeper each second.

47. 4.3 Acceleration on position-time graphs
When a car is slowing down, the speed decreases so the car covers less distance each second.
The position vs. time graph gets shallower with time.

48. 4.3 Free fall
An object is in free fall if it is accelerating due to the force of gravity and no other forces are acting on it.

49. 4.3 Acceleration and direction
A car driving around a curve at a constant speed is accelerating because its direction is changing.

50. 4.3 Curved motion A soccer ball is an example of a projectile.
A projectile is an object moving under the influence of only gravity.
The path of the ball makes a bowl-shaped curve called a parabola.

51. 4.3 Curved motion Circular motion is another type of curved motion.
An object in circular motion has a velocity vector that constantly changes direction.

52. 4.3 Curved motion Circular motion is another type of curved motion.
An object in circular motion has a velocity vector that constantly changes direction.