Find the average velocity of a moving object in a particular time interval along a given direction if it moves first 10 metres in 5 seconds and next 20 metres in 9 seconds.
Answers
Answer:
Basic Relationships
Recall from our study of derivatives that if
x
(
t
)
is the position of an object moving along a straight line at time
t
,
then the velocity of the object is
v
(
t
)
=
d
x
d
t
,
and the acceleration is given by
a
(
t
)
=
d
v
d
t
=
d
2
x
d
t
2
.
Using the integral calculus, we can calculate the velocity function from the acceleration function, and the position function from the velocity function.
The Fundamental Theorem of Calculus says that
t
2
∫
t
1
a
(
t
)
d
t
=
v
(
t
)
|
t
2
t
1
=
v
(
t
2
)
−
v
(
t
1
)
.
Similarly, the difference between the position at time
t
1
and the position at time
t
2
is determined by the equation
t
2
∫
t
1
v
(
t
)
d
t
=
x
(
t
)
|
t
2
t
1
=
x
(
t
2
)
−
x
(
t
1
)
.
If the object moves from the position
x
(
t
1
)
to the position
x
(
t
2
)
,
the change
x
(
t
2
)
−
x
(
t
1
)
is called the displacement of the object:
Δ
x
=
x
(
t
2
)
−
x
(
t
1
)
.
To find the total distance traveled by the object between time
t
1
and time
t
2
,
we need to compute the integral of
|
v
(
t
)
|
:
d
=
t
2
∫
t
1
|
v
(
t
)
|
d
t
.
Answer:
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