Kinematics

Rectilinear Motion
 Introduction

Kinematics (from the Greek kinêma, movement)

Study of motion,
Three main characteristics of the body:
Position,
Velocity,
 Acceleration.
 Introduction

Material point: to simplify the study of an

object's motion, we often focus on the motion
of its center of mass. This point, with mass m,
then replaces the object.
Trajectory: this is the geometric locus of the
positions successively occupied
by the material point over time.
Trajctory
 Introduction

Trajectory is a straight line the motion is rectilinear,

Trajectory is a circle the motion is circular,

Trajectory is a curve the motion is curvilinear.

Reference system: to describe the motion of a body, it is essential to

have a reference system (a frame of reference).

Example: two observers, one linked to the ground and the other to the
a moving car, describe differently, in their respective systems, the
movement of an object falling near a tree.
 Rectilinear motion
Rectilinear motion: Position, velocity and
acceleration :
Question: Do you know why we start by
studying the rectilinear motion?
Answer: Rectilinear motion is the simplest of all
motions, since only one component is needed to
describe it. A scalar study is therefore sufficient;
we don't need to use vectors.
 Rectilinear motion
In the case of rectilinear motion, the trajectory is a
straight line; for orientation, it is given an axis with
an origin O;
Example: axis (Ox) with unit vector 𝑖.
Position :
To locate a body, we need to know its position x on
this axis.
x represents the abscissa an algebraic quantity.
Its absolute value |x| represents the distance
separating mobile M from the origin O.
 Rectilinear motion
Application 1: (Involve students in building a
measurement table)
1. Draw a rectilinear trajectory and fit it with an
axis (Ox).
2. Show Mi points (i=1 to 7) corresponding to
increasing times of your choice; M1 corresponds
to t=0s.
3. Fill in a measurement table giving positions
and associated times.
 Rectilinear motion
Position-time (or x(t) graph) :
The values in the table can be represented on a
system of axes (O, t, x).

The various experimental points can be

connected by a curve called the x(t) graph,

Comments on the position-time graph.

 Rectilinear motion
Application 2:
The following table gives the positions of
the ball as a function of time.

t(s) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Position
0.000 0.025 0.094 0.198 0.332 0.490 0.668 0.862 1.069 1.287 1.514
(m)

Table.1. Table of positions.

 Rectilinear motion
Position-time graph or x(t) graph

𝑥 𝑚/𝑠

𝑡(𝑠)
 Rectilinear motion
Velocity :
We can further clarify motion by adding a notion that
tells us something about the state of motion: velocity.
Velocity is a quantity that characterizes the change in
position over time.
Average velocity :
Let 𝑥1 and 𝑥2 be the positions of a mobile at times 𝑡1 and
𝑡2 respectively; the average velocity is defined by :
∆𝑥 𝑥2 − 𝑥1
𝑣𝑎𝑣𝑒 = =
∆𝑡 𝑡2 − 𝑡1
Note: ∆x represents displacement.
 Rectilinear motion

𝑥 𝑚

𝑡(𝑠)

Secants for the graph x(t) with slopes given on table. 2.

 Rectilinear motion
Slopes of
secants

𝑡1 𝑠 𝑡2 𝑠 ∆𝑡 𝑠 ∆𝑥 𝑚 ∆𝑥
𝑣𝐚𝐯𝐞 = 𝑚/𝑠
∆𝑡
0.0 2.0 2.0 1.514 0.757
0.2 1.8 1.6 1.262 0.788
0.4 1.6 1.2 0.974 0.811
0.6 1.4 0.8 0.664 0.830
0.8 1.2 0.4 0.336 0.840
Table.2. Table of average velocities
for a time interval centered on
𝒕𝟎 = 𝟏𝒔.
 Rectilinear motion
• Average velocity is therefore the slope of the
secants connecting positions 𝑥2 and 𝑥1 on the
𝑥 𝑡 graph.
• Instantaneous velocity :
Average velocity gives us the valid velocity over
an interval of time ∆t,
At a given instant t instantaneous velocity.
 Rectilinear motion
As ∆t decreases, the average velocity we
calculate approaches the instantaneous
velocity at the instant corresponding to the
middle of the time interval:
𝑡1 + 𝑡2 𝑡2
𝑣 𝑡= ≈ 𝑣𝑎𝑣𝑒 𝑡
2 1
Tangent to a graph x(t)
 Rectilinear motion
There are various techniques for determining
instantaneous velocities for a given instant t. We can
operate either by :
1. equating the mean velocity calculated over a time
interval ∆t with the velocity at the center of the time
𝑡1 +𝑡2 𝑡2
interval→ 𝑣 ≈ 𝑣𝑚𝑜𝑦 𝑡 .
2 1
2. determining the slope of the tangent to the graph x(t)
at time t,
3. deriving the function 𝑥 𝑡 →
∆𝑥 𝑑𝑥
𝑣 𝑡 = lim = =𝑥
∆𝑡→0 ∆𝑡 𝑑𝑡
𝑑𝑥
Note: reads: time derivative of 𝑥 .
𝑑𝑡
 Rectilinear motion
Application :
If 𝑥 𝑡 = 5 𝑡 2 → 𝑣 𝑡 = 𝑥 = 10𝑡 where 𝑥 is
given in m, t in s and 𝑥 in m/s.
Acceleration :
Acceleration is a quantity that characterizes
the change in velocity over time.
The methodology for studying the
acceleration is the same as that used for
velocity. The same conclusions apply.
 Rectilinear motion
Average acceleration :
Let 𝑣1 and 𝑣2 be the velocities of the moving
body at times 𝑡1 and 𝑡2 respectively. Thus, we
obtain :
∆𝑣 𝑣2 − 𝑣1
𝑎𝑎𝑣𝑒 = =
∆𝑡 𝑡2 − 𝑡1
Instantaneous acceleration :
∆𝑣 𝑑𝑣 𝑑2 𝑥
𝑎 𝑡 = lim = =𝑣= 2 =𝑥
∆𝑡→0 ∆𝑡 𝑑𝑡 𝑑𝑡
 Rectilinear motion
As with velocity, acceleration is obtained by :
1. considering that the average acceleration calculated
over a time interval is the acceleration at the center
of the time interval:
𝑡1 + 𝑡2 𝑡2
𝑎 𝑡= ≈ 𝑎𝑚𝑜𝑦 𝑡
2 1
2. measuring the slopes of the tangents at the various
points on the graph v(t).
3. derivating :
𝑑𝑣 𝑑2 𝑥
𝑎 𝑡 = =𝑣= 2 =𝑥
𝑑𝑡 𝑑𝑡
 Rectilinear motion
Introduction to integral calculus :
From velocity to space - integral calculation
Problem: How to solve the inverse problem, i.e.
to find the displacement distance ∆x or the
position x knowing the velocity values.
i.e.: Can we find the positions of a moving
object once its velocities are known?
 Rectilinear motion
Case 1: 𝑣 = 𝑣𝑚 = 𝑐𝑠𝑡𝑒 → ∆𝑥 = 𝑣 ∆𝑡

1 2

∆x represents the area under the velocity curve

and it is exactly the displacement of the mobile
between the times 𝑡1 et 𝑡2 .
 Rectilinear motion
Moreover, if we also know the position 𝑥1 , it will
be possible to know the position 𝑥2 because:
∆𝑥 𝑥2 − 𝑥1
𝑣 = 𝑣𝑚 = = 𝑡ℎ𝑒𝑛
∆𝑡 𝑡2 − 𝑡1
𝐴 = ∆𝑥 = 𝑣𝑚 ∆𝑡 = 𝑥2 − 𝑥1 = 𝑣𝑚 𝑡2 − 𝑡1
And so, the position is given by :
𝑥2 = 𝑥1 + 𝑣𝑚 𝑡2 − 𝑡1 = 𝑥1 + 𝐴
 Rectilinear motion
Case 2: velocity 𝑣 varies with time.
𝑣

𝑣2

𝑣1
𝑡
𝑡1 𝑡2

It's sufficient to divide it into n small intervals. Each of these intervals

corresponds to a displacement ∆𝒙𝒊 and we obtain :
Discrete case:
𝑛
𝐴 = ∆𝑥 = 𝑥2 − 𝑥1 = 𝑖=1 ∆𝑥𝑖

Thus : 𝑥2 = 𝑥1 + ∆𝑥 = 𝑥1 + 𝐴
 Rectilinear motion
Continuous case:
Summation is then replaced by an integral
𝑥2 𝑡2
𝑥2
𝐴 = ∆𝑥 = 𝑑𝑥 = 𝑥 𝑥 = 𝑥2 − 𝑥1 = 𝑣 𝑑𝑡
𝑥1 1 𝑡1
𝑡2
𝑥2 = 𝑥1 + 𝐴 = 𝑥1 + 𝑣 𝑑𝑡
𝑡1
Application :
Knowing that the velocity is given by 𝑣 = 3𝑡 2 , find the
displacement between 𝑡0 and an instant 𝑡, then find the
position 𝑥 at a given instant t if at 𝑡0 we have 𝑥 = 𝑥0 .
 Rectilinear motion
Solution:
The displacement or space covered is given by :
𝑥 𝑡
∆𝑥 = 𝑑𝑥 = 𝑥 − 𝑥0 = 𝑣 𝑑𝑡 = 𝑡 3 − 𝑡03
𝑥0 𝑡0
Assuming 𝑥0 is known, the position is given by :
𝑥 𝑡 = 𝑡 3 − 𝑡03 + 𝑥0
 Rectilinear motion
Some primitives:
Primitive 𝑡𝑛 𝑒 𝑘𝑡 sin 𝑡 cos 𝑡 𝑡𝑔 𝑡 𝑢𝑛 𝐿𝑜𝑔 𝑢

Derivative 𝑛 𝑡 𝑛−1 𝑘𝑒 𝑘𝑡 cos 𝑡 − sin 𝑡 𝑛 𝑢𝑛−1 𝑢 𝑢 𝑢

From acceleration to velocity
The previous reasoning remains valid, and the methodology to be followed remains the
same.
Application :
Knowing that acceleration is given by 𝑎 = 2 𝑡, find the velocity 𝑣 at a given instant 𝑡.
𝑣 𝑡
𝑑𝑣
𝑎= ⇒ 𝑑𝑣 = 𝑎 𝑑𝑡 ⇒ 𝑑𝑣 = 𝑎 𝑑𝑡 ⇒ 𝑣 − 𝑣0 = 𝑡 2 − 𝑡02 ⇒ 𝑣 = 𝑡 2 − 𝑡02 + 𝑣0
𝑑𝑡 𝑣0 𝑡0
 Rectilinear motion
In short:

Analytically, we derive or we integrate

x(t)

v(t)

a(t)
 Rectilinear motion
Graphically:
From the left to the right, we integrate, but from
the right to the left, we derive:

0
 Rectilinear motion
𝒂 𝒗 gives the nature of the motion. Thus, if :
𝑎 𝑣 > 0 → Accelerated motion,
𝑎 𝑣 < 0 → Decelerated motion.
Some special rectilinear motions :
The vector expression is useless and the scalar
expression is sufficient for rectilinear motion. For
this type of motion, there are :
 Rectilinear uniform motion:
𝑎 = 0 and 𝑣 = 𝑐𝑠𝑡𝑒
 Rectilinear uniformly varied motion 𝑎 = 𝑐𝑠𝑡𝑒.
Rectilinear motion

𝒂 𝒗 x 𝒗 𝒂 x

Accelerated motion (va>0)

𝒂 𝒗 x 𝒗 𝒂 x

Decelerated motion (va<0)