2. Kinematics
a) A body is at rest relative to a certain reference, if its position doesn’t change
relative to that reference.
b) A body is in motion relative to a certain reference, if its position changes relative to
that reference.
1) Rest and Motion:
2) Trajectory of a moving object:
The trajectory of a moving object is the path followed by this object during its motion
Or (the set of occupied position)
• If the trajectory is a Straight line => The Motion is Rectilinear => eq: y = ax + b
• If the trajectory is a Curve => The Motion is Curvilinear => eg. Parabola
of eq: 𝑦𝑦 = 𝑎𝑎𝑎𝑎2
+ 𝑏𝑏𝑏𝑏 + 𝑐𝑐
• If the trajectory is a Circle => The Motion is Circular => eq: (𝑥𝑥 − 𝑎𝑎)2
+(𝑦𝑦 − 𝑏𝑏)2
= 𝑅𝑅2
where (a, b) are the coordinates of the center of the circle an R is its radius.
3. a) Frame of Reference:
To describe the motion of a body, its necessary to define a frame of reference (since
motion is relative). A suitable frame of reference is the Cartesian coordinate system (we
will use only two axis X and Y) XOY, that composes of two perpendicular axis 𝑂𝑂𝑂𝑂 and 𝑂𝑂𝑌𝑌
with unit vectors ⃗𝚤𝚤 and ⃗𝚥𝚥 respectively and origin O (0, 0)
b) Time Reference:
• Initial Time: 𝑡𝑡0 = 0
is the instant at which we start timing (it’s not necessary the instant we start the
motion)
• Instant and Duration:
The instant t at which an event occurs is determined relative to the initial time
The Duration ∆t is how long an event lasts from the beginning (initial) to the end
(final) ∆𝒕𝒕 = 𝑡𝑡𝑓𝑓 − 𝑡𝑡𝑖𝑖
3) References:
4. Consider a particle M moving on a
plane, the position vector of M at an
instant t is 𝑂𝑂𝑂𝑂 = ⃗𝑟𝑟 = 𝑥𝑥⃗𝚤𝚤 + 𝑦𝑦⃗𝚥𝚥
Where x and y are the coordinates of
M at the instant t.
The position vector ⃗𝑟𝑟 has 4
characteristics:
Origin: O
Line of action: OM
Direction: from O to M
Magnitude: 𝑂𝑂𝑂𝑂 = 𝑥𝑥2 + 𝑦𝑦2
Example: ⃗𝑟𝑟 = 3𝑡𝑡⃗𝚤𝚤 + 𝑡𝑡2⃗𝚥𝚥
At t = 2sec ⃗𝑟𝑟2 = 6⃗𝚤𝚤 + 4⃗𝚥𝚥
i.e the particle exists at a point with
coordinates x = 6 and y = 4 at the
instant 2 seconds.
4) Position vector:
6. Vector and Scalar
A vector quantity is characterized by having both a
magnitude and a direction.
Displacement, Velocity, Acceleration, Force …
Denoted in boldface type v, a, F… or with an arrow over the top
⃗𝑣𝑣, ⃗𝑎𝑎, ⃗𝐹𝐹 … . .
A scalar quantity has magnitude, but no direction.
Distance, Mass, Temperature, Time …
For motion along a straight line, the direction is
represented simply by + and – signs.
+ sign: Right or Up.
− sign: Left or Down.
7. a) Distance:
The distance (d) covered by an object is the length of the followed path
b) Displacement vector:
The displacement vector ∆𝒓𝒓 of a moving particle, between 𝑡𝑡1 𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡2 is a vector that
joins the positions of the particle at 𝑡𝑡1 𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡2.
𝒓𝒓𝟏𝟏 + ∆𝒓𝒓 = 𝒓𝒓𝟐𝟐 =>
∆𝒓𝒓 = 𝒓𝒓𝟐𝟐 − 𝒓𝒓𝟏𝟏 = 𝑨𝑨𝑨𝑨 or
∆𝒓𝒓 = ∆𝑥𝑥⃗𝒊𝒊 + ∆𝑦𝑦⃗𝒋𝒋 where ∆𝒙𝒙 = 𝑥𝑥𝑓𝑓 − 𝑥𝑥𝑖𝑖 and ∆𝒚𝒚 = 𝑦𝑦𝑓𝑓 − 𝑦𝑦𝑖𝑖
Note: ∆⃗𝑟𝑟 ≠ �𝐴𝐴𝐴𝐴
The characteristics of the displacement vector ∆⃗𝑟𝑟 are :
Origin: the initial position of the particle at 𝑡𝑡1
Line of action: the line joining the initial to the final position
Direction: from the initial to the final position
Magnitude: ∆⃗𝑟𝑟 = (∆𝑥𝑥)2 + (∆𝑦𝑦)2
5) Distance and Displacement:
8. a) Average speed:
The average speed of the particle M moving from A to B is the distance (d) divided by
the time interval ∆t.
𝑉𝑉𝑎𝑎𝑎𝑎 =
�𝐴𝐴𝐴𝐴
∆𝑡𝑡
=
𝑑𝑑
∆𝑡𝑡
b) Average velocity vector:
Is the displacement vector ∆𝒓𝒓 divided by the time interval ∆t.
𝑉𝑉𝑎𝑎𝑎𝑎 =
∆𝒓𝒓
∆𝑡𝑡
=
𝒓𝒓𝟐𝟐 − 𝒓𝒓𝟏𝟏
𝑡𝑡2 − 𝑡𝑡1
Magnitude of 𝑉𝑉𝑎𝑎𝑎𝑎: 𝑉𝑉𝑎𝑎𝑎𝑎 =
∆𝒓𝒓
∆𝑡𝑡
Note: ⃗𝑉𝑉𝑎𝑎𝑎𝑎 ≠ 𝑉𝑉𝑎𝑎𝑎𝑎
6) Average speed and Average velocity vector:
9. ⃗𝑎𝑎 =
𝒅𝒅𝒗𝒗
𝑑𝑑𝑡𝑡
= 𝑣𝑣𝑣
Origin: 𝑀𝑀𝑖𝑖
Magnitude: ⃗𝑎𝑎 = ⃗𝑎𝑎
7) Instantaneous Acceleration vector:
8) Position of the center of mass of a system of particles:
𝑂𝑂𝑂𝑂 = ⃗𝑟𝑟𝐺𝐺 =
𝑚𝑚1 ⃗𝑟𝑟1 + 𝑚𝑚2 ⃗𝑟𝑟2 + 𝑚𝑚3 ⃗𝑟𝑟3 + 𝑚𝑚4 ⃗𝑟𝑟4 … … … + 𝑚𝑚𝑛𝑛 ⃗𝑟𝑟𝑛𝑛
𝑚𝑚1 + 𝑚𝑚2 + 𝑚𝑚3 + 𝑚𝑚4 … … . . +𝑚𝑚𝑛𝑛
10. Application 3
The expression of the position vector of a moving particle M is:
a) ⃗𝑟𝑟 = 2𝑡𝑡⃗𝚤𝚤 + (4𝑡𝑡2 − 8𝑡𝑡 + 6)⃗𝚥𝚥
b) ⃗𝑟𝑟 = 3 cos(3𝑡𝑡) ⃗𝚤𝚤 + 3 sin(3𝑡𝑡) ⃗𝚥𝚥
1. Determine the shape of the trajectory of M in each of the above cases.
2. Determine the displacement vector between 𝑡𝑡1 = 1 sec 𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡2 = 3𝑠𝑠𝑠𝑠𝑠𝑠. for part
(a) only.
3. Deduce the average velocity vector between 𝑡𝑡1 = 1 sec 𝑎𝑎𝑎𝑎𝑎𝑎 𝑡𝑡2 = 3𝑠𝑠𝑠𝑠𝑠𝑠. for part
(a) only.
4. Determine the velocity and acceleration vectors for M at any instant in each
of the above cases.
Note: to determine the shape of the trajectory, you should determine first the
trajectory equation i.e the relation between x and y independent on t.
So, you should put first the parametric equations i.e x(t) and y(t)
Assignment
11. Application 4
Consider three particles:
(A) Of mass 𝑚𝑚1 = 2 𝑘𝑘𝑘𝑘 and position vector ⃗𝑟𝑟1 = 𝑡𝑡2
⃗𝚤𝚤 + 15𝑡𝑡⃗𝚥𝚥
(B) Of mass 𝑚𝑚2 = 2 𝑘𝑘𝑘𝑘 and position vector ⃗𝑟𝑟2 = 5𝑡𝑡⃗𝚤𝚤 − 10𝑡𝑡⃗𝚥𝚥
(C) Of mass 𝑚𝑚3 = 1 𝑘𝑘𝑘𝑘 and position vector ⃗𝑟𝑟3 = −2𝑡𝑡2⃗𝚤𝚤 + 5⃗𝚥𝚥
1. Determine the position vector ⃗𝑟𝑟𝐺𝐺 of the center of mass G of the system of the
three particles.
2. Deduce the velocity vector 𝑉𝑉𝐺𝐺 of the center of mass G.
