The following are statements regarding distance-time graphs.
I. The steeper the graphs, the faster the motions.
II. Horizontal lines means the object is not changing its position(not moving).
III. Horizontal lines means the object is moving on a plain surface.
IV. Downward sloping line means the object is returning to the initial position.
Which of these pair of statements is correct regarding these graphs?
E. I and IV.
F. II and III.
G I and III.
H III and IV.
Answers
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Explanation:
How are position vs. time graphs useful?
Many people feel about graphs the same way they do about going to the dentist: a vague sense of anxiety and a strong desire for the experience to be over with as quickly as possible. But position graphs can be beautiful, and they are an efficient way of visually representing a vast amount of information about the motion of an object in a conveniently small space.
What does the vertical axis represent on a position graph?
The vertical axis represents the position of the object. For example, if you read the value of the graph below at a particular time you will get the position of the object in meters.
Try sliding the dot horizontally on the graph below to choose different times and see how the position changes.
Concept check: What is the position of the object at time t=5t=5t, equals, 5 seconds according to the graph above?
Show me the answer.
What does the slope represent on a position graph?
The slope of a position graph represents the velocity of the object. So the value of the slope at a particular time represents the velocity of the object at that instant.
To see why, consider the slope of the position vs. time graph shown below: Wait, why is the vertical axis called x?
The slope of this position graph is \text{slope}=\dfrac{\text{rise}}{\text{run}}=\dfrac{x_2-x_1}{t_2-t_1}slope=
run
rise
=
t
2
−t
1
x
2
−x
1
start text, s, l, o, p, e, end text, equals, start fraction, start text, r, i, s, e, end text, divided by, start text, r, u, n, end text, end fraction, equals, start fraction, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, divided by, t, start subscript, 2, end subscript, minus, t, start subscript, 1, end subscript, end fraction.
This expression for slope is the same as the definition of velocity: v=\dfrac{\Delta x}{\Delta t}=\dfrac{x_2-x_1}{t_2-t_1}v=
Δt
Δx
=
t
2
−t
1
x
2
−x
1
v, equals, start fraction, delta, x, divided by, delta, t, end fraction, equals, start fraction, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, divided by, t, start subscript, 2, end subscript, minus, t, start subscript, 1, end subscript, end fraction. So the slope of a position graph has to equal the velocity.
This is also true for a position graph where the slope is changing. For the example graph of position vs. time below, the red line shows you the slope at a particular time. Try sliding the dot below horizontally to see what the slope of the graph looks like for particular moments in time.