Calculus Of One Real Variable – By Pheng Kim Ving

12.5 
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1. Distance Travelled And Displacement 
We're now going to find the distance travelled by and the
displacement of an object moving on a straight line given that
its velocity v = v(t) at any time t is known.
Fig. 1.1 Distance Travelled And Displacement. 
Let's distinguish between distance travelled and
displacement. Suppose the motion is along the xaxis. See Fig. 1.1. For
example:
Motion 
Distance Travelled 
Displacement 
If object moves from point 1 to point 3 
2 
2 
If object moves from point 1 to point 3 then in reverse to point 2 
3 
1 
If object moves from point 1 to point 3 then in reverse to point 1 
4 
0 
If object moves from point 1 to point 3 then in reverse to point 0 
5 
–1 

Note that distance travelled is a nonnegative quantity
while displacement is a signed quantity. The object's displacement
is positive, respectively negative, if its final position is to the right,
respectively to the left, of its initial position. Displacement
may or may not be equal to distance travelled.
Distance Between 2 Points. Refer to Fig. 1.1. The
distance between point 1 and point 3 is 2. If the object travels from
point 1 to point 3, then its distance travelled is 2. If the object travels
from point 1 to point 3, reverses direction and
travels to point 2, and reverses direction and travels to point 3, then its distance
travelled is 2 + 1 + 1 = 4. The distance
between 2 points may or may not equal the total distance travelled by an object
between them. It equals the absolute
value of the displacement of the object between them.
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2. Finding Distance Travelled And Displacement 
Recall from Section
5.8 that speed is the absolute value of velocity, and that velocity is
positive, respectively negative, if
the object moves in the positive, respectively negative, direction. On the
normal xaxis, the
positive, respectively negative,
direction is lefttoright, respectively righttoleft.
Fig. 2.1

Fig. 2.2
Velocity v = k < 0,
where k is a constant.

Fig. 2.3
Motion Along xAxis In Positive Direction With Variable Velocity.

Fig. 2.4
Distance Travelled = Displacement = Area
Of Colored Region.

Fig. 2.5
Motion Along xAxis In Negative Direction With Variable Velocity.

Fig. 2.6
Distance Travelled = Area Of Colored
Region;

Fig. 2.7
Motion Along xAxis In Positive And Negative Directions With Variable

Fig. 2.8
Distance Travelled = A_{1} + A_{2};

In summary:

Looking at Fig. 2.7 you may wonder what about x_{2} if we integrate from x_{1} to x_{3}, as x_{2} lies outside
the interval [x_{1}, x_{3}],
so it appears that the motion from x_{2} to x_{3} is excluded. Well, we don't
integrate from x_{1} to x_{3}, we integrate the velocity
v(t) from t_{1} to t_{3}, and t_{2} lies inside the interval [t_{1}, t_{3}], as seen in
Figs. 2.7 and 2.8, so the velocity from t_{2} to t_{3} and thus
the motion from x_{2} to x_{3} are included.
a. The distance travelled by the object.
b. The displacement of the object.
Solution
a. Distance travelled:
b. Displacement:
EOS
Problems & Solutions 
a. The distance
travelled by the object.
b. The displacement of the
object.
Solution
a. v(0) = cos 0 = 1; so at time t = 0 sec the object travels in the positive direction with a speed of 1 m/sec.
Thus:
At time t = 3 sec, the object is back at where it was at time t = 0 sec.
a. The
distance travelled by the object.
b. The displacement of the
object.
a. Clearly:
from above calculation.
4. A body moves on the xaxis with acceleration a(t)
= d^{2}x/dt^{2} = 6t
m/sec^{2}. It starts at time t
= 0 with initial velocity
v_{0} = –3 m/sec.
a. Find the velocity v(t) as a function of t.
b. Find the total distance s travelled by the body from
time t = 0 sec
to time t = 4 sec.
c. Where is its position at
time t = 4 sec
relative to its position at time t
= 0 sec?
Solution
s = 56 m.
c. Displacement:
At time t = 4 sec, the position of the body is 52 m to the right of its position at time t = 0 sec.
v(t) = 2e^{–}^{t}
= 2/e^{t}
> 0 for all t,
so speed = v(t) = v(t) = 2/e^{t};
speed decreases very rapidly toward 0 but is always > 0
as time passes.
Solution
The object moves 2 km throughout eternity.
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