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Graphs of Linear Motion

Graphs of Linear Motion

Graphs of distance (position) versus time are commonly used to represent the linear motion of an object.

o X-axis (independent variable) – time

o Y-axis (dependent variable) – distance, position, displacement

o Slope – speed (velocity)

You try: To begin, consider a car moving with a constant, rightward (+) velocity of 25 m/s.

[pic]

|[pic] |Sketch the graphs of motion indicated for the car |[pic] |

| |shown above. | |

1. Calculate the slope of the position-time graph. What is its physical significance?

[pic]

2. Calculate the area beneath the velocity-time graph. What is its physical significance?

[pic]

Using the graph to the left, determine which runner is moving with the greatest speed. Explain.

Runner A, because it has the steepest

slope.

Consider a car moving with a constant velocity of +5 m/s for 5 seconds, stopping abruptly, remaining at rest (v = 0 m/s) for 5 seconds and then returning back to its original position.

[pic]

Now, sketch a position time graph representing the velocity of the car.

[pic] [pic]

Graphs of Changing Velocity

Now consider a car moving with a changing, rightward (+) velocity – that is, a car that is moving rightward and speeding up or accelerating.

[pic]

Sketch a position-time graph for the cars above.

[pic]

Sketch the graphs below accordingly.

|[pic] |[pic] |[pic] |

|Time |Time |Time |

|Constant speed in positive direction |Constant speed in negative direction |Standing still |

Graphs of Accelerated Linear Motion

Consider a car with a rightward (+) changing velocity. That is, a car moving to the right and speeding up or accelerating. Since the car is moving in the positive direction and speeding up, it is said to have a positive acceleration.

[pic]

|Time (s) |0 - 1 |1 - 2 |2 - 3 |3 - 4 |4 - 5 |

|Distance (m) |2 |6 |10 |14 |18 |

|Velocity (m/s) |4 |8 |12 |16 |20 |

|Acceleration |4 m/s2 (speed increased 4 m/s each second) |

Sketch a velocity-time graph for the cars above.

|[pic] | |

| |What does the slope of the graph represent? |

| | |

| |acceleration |

| | |

| |Calculate the area underneath the slope. What is its physical significance? |

| |distance, [pic] |

Characteristics of Velocity vs. Time graphs

o X-axis (independent variable) – Time

o Y-axis (dependent variable) – Velocity

o Slope - Acceleration

• If slope is zero, than acceleration is _______zero_________.

• If slope is positive, than acceleration is _______positive_____.

• If slope is negative, than acceleration is _______negative_____.

How can you tell if the object is moving in the positive direction (i.e., positive velocity) or in the negative direction (i.e., negative velocity)?

Positive Velocity Negative Velocity

[pic] [pic][pic][pic]

How can you tell if the object is speeding up or slowing down?

Speeding Up Slowing Down

[pic] [pic]

Sketch the graphs below accordingly.

|[pic] |[pic] |[pic] |

|Time |Time |Time |

|Increasing speed |Decreasing speed |Constant speed, then decreasing speed to stop. |

Consider the graph below. The object whose motion is represented by this graph is ... (circle all that apply):

a. moving in the positive direction.

b. moving with a constant velocity.

c. moving with a negative velocity.

d. slowing down.

e. changing directions.

f. speeding up.

g. moving with a positive acceleration.

h. moving with a constant acceleration.

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How can you tell that the car is moving at a constant speed?

[pic]

The slope is the velocity.

The area is the distance traveled.

Time (s)

Distance (m)

To linearize graph, plot (time)2 on the x-axis.

Position

Position

Position

Equations needed to solve for velocity will be learned at a later date.

Velocity values are for the end of the time interval.

backwards

forewards

+ away

- towards

Position

Position

+ away

- towards

Position

How can you tell that the car is speeding up?

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