Calculus Of One Real Variable – By Pheng Kim Ving
Chapter 12: Applications Of The Integral – Section 12.5: Distance And Displacement

12.5
Distance And Displacement

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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 x-axis. 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 non-negative 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 x-axis, the positive, respectively negative,
direction is left-to-right, respectively right-to-left.

Constant Velocity

Fig. 2.1

Fig. 2.2

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

Variable Velocity

Fig. 2.3

Motion Along x-Axis In Positive Direction With Variable Velocity.

Fig. 2.4

Distance Travelled = Displacement = Area Of Colored Region.

Fig. 2.5

Motion Along x-Axis In Negative Direction With Variable Velocity.

Fig. 2.6

Distance Travelled = Area Of Colored Region;
Displacement = Negative Of That Area.

Fig. 2.7

Motion Along x-Axis In Positive And Negative Directions With Variable
Velocity.

Fig. 2.8

Distance Travelled = A₁ + A₂;
Displacement = A₁ – A₂.

In summary:

Looking at Fig. 2.7 you may wonder what about x₂ if we integrate from x₁ to x₃, as x₂ lies outside the interval [x₁, x₃],
so it appears that the motion from x₂ to x₃ is excluded. Well, we don't integrate from x₁ to x₃, we integrate the velocity
v(t) from t₁ to t₃, and t₂ lies inside the interval [t₁, t₃], as seen in Figs. 2.7 and 2.8, so the velocity from t₂ to t₃ and thus
the motion from x₂ to x₃ are included.

Example 2.1

a. The distance travelled by the object.
b. The displacement of the object.

Solution

a. Distance travelled:

b. Displacement:

EOS

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Problems & Solutions

a. The distance travelled by the object.
b. The displacement of the object.

Solution

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.

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a. The distance travelled by the object.
b. The displacement of the object.

Solution

a. Clearly:

from above calculation.

4. A body moves on the x-axis with acceleration a(t) = d²x/dt² = 6t m/sec². It starts at time t = 0 with initial velocity
    v₀ = –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?