INTRODUCTION TO DYNAMICS

Dynamics is the branch of science which deals with the study of behavior of body or particle in the
state of motion under the action of force system. The first significant contribution to dynamics was
made by Galileo in 1564. Later, Newton formulated the fundamental laws of motion.
Dynamics branches into two streams called kinematics and kinetics.
Kinematics is the study of relationship between displacement, velocity, acceleration and time of the
given motion without considering the forces that causes the motion.

Kinetics is the study of the relationships between the forces acting on the body, the mass of the body
and the motion of body,

Motion: A body is said to be in motion, if it is changing its position with respect to a reference point.

Particle: A particle is defined as a material point without dimensions but containing a definite
quantity of matter.

Path : It is the imaginary line connecting the position of a body or particle that has been occupied at
different instances over a period of time . This path. Traced by a body or particle can be a straight
line or curve

Rectilinear motion: Whenever a particle moves along a straight line, and the particle is said to have
rectilinear motion.

Displacement and Distance Travelled

Distance is a scalar quantity, measure of the interval between two locations measured along the
actual path connecting them. Distance is an absolute quantity and always positive.

Displacement is a vector quantity, It is measure of the length covered between two points in a
particular direction, measured along the shortest path connecting them. SI unit is meter.

Taking the line of motion along the x-axis (Fig.1),

we can define the displacement of the particle by its
x-coordinate, measured from the fixed reference
point O. We shall consider this displacement as
positive when the particle is to the right of the
origin O and as negative when to the left.
Fig.1 line of motion along X-axis
 As the particle moves, the displacement varies with time, and the motion of the particle is
completely defined if we know the displacement x at any instant of time t , this relation can be
expressed by displacement time equation

x = ƒ (t)------(1) where ƒ (t) stands for any function of time.

Depending upon how the particle moves along the x-axis, Eq.(1)will take different forms.

Case1: uniform rectilinear motion, the displacement x is represented by

x = c + b t ------ (1a) where c is constant which represents an initial displacement at the time t = O,

b is constant which represents the rate at which the displacement increases.

Case 2 : For freely falling bodies, the displacement x is represented by

x = (1/2) gt2 -------------- (1b)

Case 3: The motion, in which the displacement decreases exponentially with time,

X = re-kt ------------ (1c)

Where r and k are constants and e is the natural logarithmic base.

Example: a particle projected into a highly viscous medium which finally bring it to rest.

Fig-2 Displacement –Time graph

Instead of analytical expressions for Eq. (1), it is often useful to represent the displacement-time
relationship graphically. Taking time t as abscissa(x-axis) and displacement x as ordinate(y-axis), the
curve represented by Eq. (1) can be plotted for any particular case. This gives us a so-called
displacement-time diagram.. These curves represent graphically the same information as given by the
analytic expressions.
Speed: Rate of change of distance travelled by the particle with respect to time is called speed.

Velocity: Rate of change of displacement with respect to time is called velocity

denoted by v. SI unit is m

Mathematically v = dx/dt (2)

Average velocity: When an object undergoes change in velocities at different instances, the average
velocity is given by the sum of the velocities at different instances divided by the number of
instances. That is, if an object has different velocities V1, V2,V3,.: , Vn, at times t = t1, t2, t3, ...,tn,
then the average velocity is given by

V = (VI+V2+V3+.... vn)/ n

𝒙𝟐 −𝒙𝟏 ∆𝒙
= = (3)
𝒕𝟐 −𝒕𝟏 ∆𝒕

Instantaneous velocity: It is the velocity of moving particle at a certain instant of

time. The instantaneous velocity of the particle for the moment ‘t’
∆𝒙 𝒅𝒙
Instantaneous velocity 𝒗 = 𝐥𝐢𝐦 = = 𝒙̇ (4)
∆𝒕→𝟎 ∆𝒕 𝒅𝒕

For the particular cases of rectilinear motion of a particle that we have represented by Eqs (1 a), (1
b), and (1c), we obtain by Eq.(4).

𝑑
𝑣= 𝑑𝑡
( c + b t) = b --------4a

𝑑
𝑣= (𝑐𝑡 2 ) = 2𝑐𝑡 --------4b
𝑑𝑡

𝑑
𝑣= 𝑑𝑡
(𝑟𝑒 −𝑘𝑡 ) = −𝑟𝑘𝑒 −𝑘𝑡 --------4c

Variations in velocity with respect to time can also be represented graphically. Taking velocity ν as
ordinate and time t as abscissa, we may plot from Eq. (4) the so-called velocity-time diagrams.

In the case of Eq. (4a),we see that the velocity-time curve is a horizontal straight line indicating
motion with constant velocity b. From Eq. (4b) we obtain, for the velocity-time diagram, a straight
line with constant slope 2C. In the final case, Eq.(4c) gives a more general velocity-time curve in
which the velocity varies exponentially with time. Velocity-time diagrams such as those shown in
Fig . are especially helpful in studying various cases of rectilinear motion.
 If we have the velocity-time diagram for any rectilinear motion of a particle, the distance travelled
by the particle during any interval of time from t = t1 to t = t2 will be represented by the area under
the curve between these two ordinates of time. To prove this statement, we consider the velocity-
time diagram shown in Fig.5.

From Eq.(4), we may write dx = ν dt

which expresses the small increment of displacement through which the particle moves during the
time interval dt.

From the figure we see that this expression also

represents the area of the infinitesimal strip of width (dt)
and height v of the velocity-time diagram. Summing up
all such increments of displacement between the instants
t1 and t2, the large shaded area is obtained as the total
distance travelled by the particle during the specified
time interval.

Thus we see that the velocity-time diagram is particularly valuable in that from which we can obtain
not only the velocity of motion at any instant but also the distance travelled during any given interval
of time.

Acceleration: Rate of change of velocity with respect to time is called acceleration

Mathematically a = dv/dt

If the particle receives equal increments of velocity Δν in equal intervals of time Δt. Then particle is
moving with constant acceleration’ a ‘given by the equation,
∆𝒗 𝒎
𝒂= 𝒂𝒏𝒅 𝑺. 𝑰 𝒖𝒏𝒊𝒕𝒔 𝒂𝒓𝒆 ( 𝒔𝟐 )
∆𝒕

∆𝐯
Average acceleration 𝐚𝐚𝐯𝐞 = ∆𝐭

∆𝒗 𝒅𝒗 𝒅𝟐 𝒙
Instantaneous acceleration 𝒂 = 𝒍𝒊𝒎 = = = 𝒙̈
∆𝒕→𝟎 ∆𝒕 𝒅𝒕 𝒅𝒕𝟐

we distinguish not only the magnitude of acceleration but also its direction. Acceleration is
considered positive when the velocity obtains positive increments in successive intervals of time and
negative if the velocity is decreasing and negative acceleration is called deceleration.
In case of recliner motion of a particle, the acceleration- 𝑑2
𝑎= (𝑐 + 𝑏𝑡) = 0
time relationship can be expressed analytically by taking 𝑑𝑡 2
the second derivative with respect to time of the
𝑑2
displacement-time Eq. (1). 𝑎= (𝑐𝑡 2 ) = 2𝑐
𝑑𝑡 2
For the three cases under consideration are represented 𝑑2
graphically by the slopes of the velocity-time diagrams 𝑎= (𝑟𝑒 −𝑘𝑡 ) = 𝑟𝑘 2 𝑒 −𝑘𝑡
𝑑𝑡 2
shown in Fig. 4.

Principle of Dynamics:

First Law: Everybody continues in its state of rest or of uniform motion in a straight line until and
unless it is compelled by force to change that state, This Law, sometimes called the inertia Law.

Second Law: The acceleration of a given particle is proportional to the force applied to it and takes
place in the direction of the straight line in which the force acts.

Third Law:-For every action there is equal and opposite reaction or the mutual actions of any two
bodies are always equal and oppositely directed.

General Equation of motion of a particle:

By using the second law of dynamics we can establish the general equation of motion of a
particle.

We know from Galileo’s experiments that the gravity force W, if acting alone, produces an
acceleration of the particle equal to g .If instead of the force W, a force F acts on the same particle,
then from the second law it follows that the acceleration produced by this force is in the direction of
the force

Acceleration due to gravity’ g ‘α W and Acceleration a α F ;

𝑎 𝐹 𝑤
=𝑊 => × 𝑎 = 𝐹 --------(1)
𝑔 𝑔

𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛(1)𝑖𝑠 𝑡ℎ𝑒 𝑔𝑒𝑛𝑒𝑟𝑎𝑙 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑚𝑜𝑡𝑖𝑜𝑛.

It is seen from eqn.(1 ) that for a given magnitude of the force F, the acceleration produced is
inversely proportional to the factor (w/g) , it is a measure of the inertia of the particle and called the
mass of the particle and is generally denoted by m.
𝑤
𝑚𝑎𝑠𝑠 𝑚 = 𝑔

The general equation of motion of a particle becomes ma = F

Differential equation of rectilinear motion.

Whenever a body moves under the action of a force applied at its center of gravity and
having a fixed line of action, acceleration of a body is produced in the same direction, we can obtain
differential equation of rectilinear motions
𝑊
𝐹= 𝑥̈
𝑔

𝑊 𝑑𝑉
𝐹= ( 𝑑𝑡 )
𝑔

𝑊 𝑑 𝑑𝑥
𝐹= (𝑑𝑡 ( 𝑑𝑡 ))
𝑔

D’ Alembert’s principle:

D’ Alembert was the first to point out that equations of motion could be written as equilibrium
equation simply by introducing inertia force in addition to the real force acting on a system, This
idea is known as D’ Alembert’s principle.

The differential equation of rectilinear motion of a particle can be written as

𝐹 = 𝑚𝑥̈

𝐹 − 𝑚𝑥̈ = 0

This equation of motion of a particle is of the same form as an equation of static equilibrium
and may be consider as an equation of dynamic equilibrium. Here F is called real force , 𝑚𝑥̈ is
inertia force equal to product of mass and its acceleration and directed oppositely to the acceleration.

−𝑊
− ∑ 𝑚𝑥̈ = − 𝑥̈ ∑ 𝑚 = 𝑥̈
𝑔

𝑊
∑ 𝐹𝑖 + (− 𝑥̈ ) = 0
𝑔
𝑊
Where W is the weight of the body. (− 𝑔 𝑥̈ ) is inertia force of the particles which acts through center
of gravity.

Advantage of D’ Alembert principle:

Instead of writing as many equations of motion as there are particles, we need to write only one
equation of dynamic equilibrium. In this way, we avoid consideration of all internal forces as well as
reactions exerted by ideal constraints. This method is very useful in the solution of engineering
problems connected with case of virtual work.
Momentum and Impulse:

The differential equation of rectilinear motion of a particle may be written in the form

𝑊 𝑑𝑥̇
𝐹= ( )
𝑔 𝑑𝑡

𝑑 𝑊
𝐹= ( 𝑥̇ )
𝑑𝑡 𝑔
𝑊
𝐹𝑑𝑡 = 𝑑 ( 𝑔 𝑥̇ ) ----- (1)

𝑊
̇ ----- (2)
𝑑 ( ) 𝑥̇ = 𝐹𝑑𝑡
𝑔

𝑊
The expression( 𝑔 ) 𝑥̇ on the left side of eq 2 is called the momentum of the particle,

By integrating eqn ( 2 ) we obtain

𝑊 𝑡
( 𝑔 ) 𝑥̇ + 𝑐 = ∫0 𝐹𝑑𝑡---------- (3)

In which the constant of integration’ c’ can be evaluated from the initial condition of the motion.
Assuming that at the initial moment, t=o the particle has the velocity 𝑥0̇ directed along the x-axis,
𝑊
we find from. Eqn(3) that 𝑐 = − ( 𝑔 ) 𝑥̇ 0

𝑡
𝑊 𝑊
( ) 𝑥̇ − ( ) 𝑥̇ 0 = ∫ 𝐹𝑑𝑡
𝑔 𝑔 0

Thus the total change in momentum of a particle moving for a finite interval of time is equal to the
impulse of the acting force during the same interval.

From above equation, the F is known as a function of time and it can be represented by a Force-time
diagram.

The right- hand side of eq2 is

represented by the area of the
shaded elemental strip of height F
and width dt in the force-time
diagram. This quantity is called the
impulse of the force F in the time
dt.
 The equation of momentum and impulse is particular useful when we are dealing with a
system of particles.

For example: let us consider the case of a gun and shell as shown in fig.

During the extremely short interval of explosion, the forces F acting on the shell and gun. since the
forces F are in the nature of action and reaction between the shell and gun, they must be at all times
be equal and opposite ,and hence their impulses for the interval of explosion are equal and opposite,
since the forces act exactly the same time t.

It w1 and w2 are the weight of the shell and gun respectively ,we find assuming the initial
velocities to be zero ,Then impulse acting on shell and gun are
𝑤1
For shell is 𝑣1 = ∫ 𝐹 𝑑𝑡
𝑔

𝑤2
For gun is 𝑣2 = ∫ 𝐹 𝑑𝑡
𝑔

𝑤1 𝑤2
Then for the entire system 𝑣1 = 𝑣2
𝑔 𝑔

𝑣1 𝑤
 = 𝑤2
𝑣2 1

It is observed that the velocities of the shell and gun after discharge are in opposite direction and
inversely proportional to the corresponding weights.

Work and Energy

Work energy theorem states that the total change in kinetic energy of a particle during a
displacement is equal to the work done by the acting force to bring the displacement

Writing the differential equation of rectilinear motion of a particle in the form

𝑤 𝑑𝑥̇
= F----------- (1)
𝑔 𝑑𝑡

And multiplying both sides by dx, we obtain

𝑤 𝑑𝑥
𝑑𝑥̇ = 𝐹 𝑑𝑥 ---------- (2)
𝑔 𝑑𝑡

𝑤 𝑥̇ 2
or 𝑑 (𝑔 ) = 𝐹 𝑑𝑥 ------- (3)
2

Integrating eq. (3) we find

 𝑤 𝑥̇ 2
(𝑔 + 𝑐) = ∫ 𝐹 𝑑𝑥 -------4
2

C is constant and can be evaluated from initial conditions of the motion. Assuming that, When the
displacement x=xo, the particle has the velocity 𝑥̇ = 𝑥̇ 0 and substituting these condition in eq 4

𝑤 𝑥0̇ 2 𝑥
𝐶 = (− 𝑔 ) + ∫0 0 𝐹 𝑑𝑥
2

𝑤 𝑥̇ 2 𝑤 𝑥0̇ 2 𝑥
(𝑔 −𝑔 = ∫𝑥 𝐹 𝑑𝑥) ------- (5)
2 2 0

This is called equation of work and energy or work energy theorem.

From, above equation, the force F is known as the function of the displacement ‘x’ of the particle as
represented by the force displacement diagram.

The right side of eq(3 )is represented by the area of the elemental strip of height F and width dx of
this diagram. This quantity represents the ‘work done’ by the force F on the infinitesimal
displacement dx, and the expression in the parentheses on the left side of Eq( 3) is called the kinetic
energy of the particle.

This equation is especially use full in cases where the actins force is a function of displacement and
velocity of particle as a function of displacement.

For Example: A body of weight W falling from a height h strikes the ground. In this case the acting
Force F = w and the total work is wh. Thus it the body starts from rest, the initial velocity 𝑥̇ 0 = 0 and
eqn (5)

𝑤 𝑣2
= 𝑤ℎ
𝑔 2

𝑣 = √2𝑔ℎ--------- (6)
Impact:

The phenomenon of collision of two moving bodies is called impact. In impact action and reaction
forces of very large magnitude acts for a very short interval of time. The magnitudes of the forces
and the duration of impact depend on the shapes of the bodies, their velocities and their elastic
properties

Let us consider the case of impact of two spheres of weights w1 and w2 having, velocities V1 and V2
respectively before impact, we assume that these velocities are directed along the line joining the
centers of the two spheres and consider them positive if they are in the positive direction of the X-
axis, This is called direct central Impact.

When V1˃V2, impact occurs, as shown in fig.

During impact, two equal and opposite forces, action and reaction, are produced at the point of
contact. And if V’1 and V’2 are the velocity of the balls after the impact, In accordance with the
law of conservation of momentum,
𝑤1 𝑤2 𝑤1 ′ 𝑤2 ′
𝑣1 + 𝑣2 = 𝑣 + 𝑣
𝑔 𝑔 𝑔 1 𝑔 2

Plastic impact:

Assume, for instance, that the material is absolutely inelastic, like pithy. Then with the beginning of
impact, plastic deformation at the point of contact begins. The velocity of the striking ball (I)
gradually diminishes owing to the reaction from the ball II and at the same time the velocity V11-V21
with which the plastic deformation at the surface of contact is progressing gradually diminishes.
Finally, when the velocity of the balls became equal V11-V21 plastic deformation ceases and owing to
the fact that the materials is absolutely in enlister.
Denoting by V1 the common velocity of the two spheres after impact, eq 1 becomes
𝑤1 𝑤2 𝑤1 𝑤2
𝑣1 + 𝑣2 = [ + ] 𝑣 ′
𝑔 𝑔 𝑔 𝑔

And we obtain far the case at plastic impact

𝑤1 𝑣1 + 𝑤2 𝑣2
𝑣′ =
𝑤1 + 𝑤2

Elastic impact:

In this case of perfect elasticity collision there must be no loss in energy of the system and we have,
in a to the momentum equation, the energy equation

𝑤1 𝑣12 𝑤2 𝑣22 𝑤1 (𝑣1′ )2 𝑤2 (𝑣2′ )2

+ = +
𝑔 2 𝑔 2 𝑔 2 𝑔 2

In which and 𝑣1′ 𝑎𝑛𝑑 𝑣2′ are the velocities of the spheres after re bound equation 5 & 1 together are
sufficient to determine the two unknown velocity and before impact and the weights W1 and W2 of
the bodies are known.

𝑤1 (𝑣1 − 𝑣1′ ) = 𝑤2 (𝑣2′ − 𝑣2 )----------------(6)

𝑤1 (𝑣12 − (𝑣1′ )2 ) = 𝑤2 ((𝑣2′ )2 − 𝑣22 )--------------(7)

(𝑣12 − (𝑣1′ )2 ) = (𝑣1 − 𝑣1′ )(𝑣1 + 𝑣1′ )

((𝑣2′ )2 − 𝑣22 ) = (𝑣2′ − 𝑣2 )(𝑣2′ + 𝑣2 )

By dividing the 7equation by 6 eq.

(𝑣1 + 𝑣1′ ) = (𝑣2′ + 𝑣2 )

(𝑣1′ − 𝑣2′ ) = −(𝑣1 − 𝑣2 )

This equation represents a combination of the law of conservation of momentum and conservation of
energy of energy. It state that for an elastic impact the relative velocity after impact has the same
magnitude as that before impact but with reversed sign.
Using this idea in conjunction with that of conservation of momentum

The equation for the case of elastic impact.

𝑤1 𝑣1 + 𝑤2 𝑣2 = 𝑤2 𝑣2′ + 𝑤1 𝑣1′

(𝑣1′ − 𝑣2′ ) = −(𝑣1 − 𝑣2 )

Let us consider different cases.

Case (i) : If W1 =W2

Before collision collision After collision

Eqn.( 8) becomes

𝑣1 + 𝑣2 = 𝑣2′ + 𝑣1′

(−𝑣1 + 𝑣2 ) = (𝑣1′ − 𝑣2′ )

Subtracting and adding these equations

= > 2 𝑣2 = 2𝑣1′ => 𝑣2 = 𝑣1′ 𝑎𝑛𝑑 𝑣1 = 𝑣2′

=>After elastic impact, equal weights simply exchange velocities.

Case (ii): If W1 =W2 and W2 is at rest before impact ie., 𝑣2 = 0

𝑣1 = 𝑣2′ + 𝑣1′

(−𝑣1 ) = (𝑣1′ − 𝑣2′ )

=> 𝑣1 = 𝑣2′ and 𝑣1′ = 0

=>The striking ball simply stops after the impact and second ball moves with the velocity of 𝑣1

Case (iii):

Before collision collision After collision

 If W1 =W2, if the two balls are moving towards each other with equal speeds v before impact ,then
they rebound with the same speed with which they collided

Case( iv) : an elastic impact of a ball against a flat immovable obstruction

In this case W2 = ∞ while W1 remains finite

and 𝑣2 = 𝑣2′ = 0 (since floor is immovable)

𝑣1′ = − 𝑣1
 Striking ball rebounds with the same speed with
which it hits the obstruction.

Ex: dropping a ball on a cement floor

Semi –elastic impact:-

In some conditions, bodies will go deviation from perfect elasticity and owing to this fact there will
be always some loss in energy of the system during impact so that the relative velocity after impact is
smaller than before and instead of equation

(𝑣1′ − 𝑣2′ ) = −(𝑣1 − 𝑣2 )

We have to take
(𝑣1′ − 𝑣2′ ) = −𝑒 (𝑣1 − 𝑣2 )

Where e is an numerical factor less than unity and called the co-efficient of restitution using this, we
have for the general case of semi-elastic directi central impact,

The following equations:

𝑤1 𝑣1 + 𝑤2 𝑣2 = 𝑤2 𝑣2′ + 𝑤1 𝑣1′

(𝑣1′ − 𝑣2′ ) = −𝑒 (𝑣1 − 𝑣2 )

It will be noted that when e=o above eq, represents the case of plastic impact. Also when
e=1,then the equation represents the case of elastic impact.
Kinematics of curvilinear motion:-

When a moving particle describes a curved path, it is said to have curvilinear motion. The path of the
particle is a plane curve.

Displacement:

To define the position of a particle P in a plane ,we need two coordinates such as x and y. These
displacements are represented as function of time.

x = f1 (t) , and y = f2(t).

when these two expressions are given ,the motion of the particle in its plane is completely defined.

The equation x = f1 (t) represents the rectilinear motion along the x- axis of the projection Px of
the particle P moving along the curved path OA.

Similarly The equation y = f2(t). represents the rectilinear motion along the y- axis of the
projection Py of the particle P moving along the curved path OA.

Thus the curvilinear motion of the particle may be considered as a compound of the two rectilinear
motions of its projections.

We can also define the motion of a particle in a plane by the equations

y = f (x) and s = f1 (t) where y = f (x) represents the equation of the path OA and s = f1 (t)
gives the displacement s measured along the path as a function of time.

Velocity: The rate of change of displacement is known as velocity of the particle.

Let us consider a small but finite interval of time from t to t+Δt during which the particle in fig 1.
Moves from p to p1 along its path. The length of chord pp1 is denoted by Δs in Δt interval of time.
∆𝑠
then average velocity ∆𝑡 = 𝑉𝑎𝑣𝑒

∆𝑥
Average velocity an x axis is (𝑣𝑎𝑣𝑒 )𝑥 = ∆𝑡

∆𝑦
Average velocity an y-axis is (𝑣𝑎𝑣𝑒 )𝑦 = ∆𝑡
The magnitude at the instantaneous velocity will be defined as follows
∆𝑠 𝑑𝑠
𝑉 = lim = 𝑑𝑡
∆𝑡→0 ∆𝑡

∆𝑥 𝑑𝑥
VX= lim = = 𝑥̇
∆𝑡→0 ∆𝑡 𝑑𝑡

∆𝑦 𝑑𝑦
VY= lim = = 𝑦̇
∆𝑡→0 ∆𝑡 𝑑𝑡

Having velocity components 𝑥̇ 𝑎𝑛𝑑𝑦̇ , 𝑤𝑒 𝑐𝑎𝑛 obtain the magnitude, direction cosines of the total
velocity v as follows:-
𝑥̇ 𝑦̇
𝑉 = √𝑥̇ 2 + 𝑦̇ 2 , cos(V,x) = cos(V,y) =
𝑉 𝑉

Acceleration:-

Consider a small interval of time from t to t + Δt during which the particle moves from p to p1 and
velocity 𝑣̅ 𝑎𝑛𝑑 𝑣̅1 at these two points on the path. Then average acceleration during the interval Δt
defined as follows.
∆𝑣
𝑎𝑎𝑣𝑒 = ∆𝑡

∆𝑣 ∆𝑥̇
Average acceleration along x-axis =(𝑎𝑎𝑣𝑒 )𝑥 = =
∆𝑡 ∆𝑡

∆𝑣 ∆𝑦̇
Average acceleration along y-axis =(𝑎𝑎𝑣𝑒 )𝑦 = =
∆𝑡 ∆𝑡

∆𝑥̇ 𝑑𝑥̇
The instantaneous acceleration along x-axis is aX= lim = = 𝑥̈
∆𝑡→0 ∆𝑡 𝑑𝑡

∆𝑦̇ 𝑑𝑦̇
instantaneous acceleration along y-axis is ay= lim = = 𝑦̈
∆𝑡→0 ∆𝑡 𝑑𝑡
 Finally the magnitude and direction cosines of the total acceleration a will be obtained from
the equation
𝑥̈ 𝑦̈
a= √𝑥̈ 2 + 𝑦̈ 2 , cos(a,x) =𝑎 cos(a,y) =𝑎

Normal and Tangential Acceleration :

Acceleration at any instant t can be resolved into another set of rectangular components
coinciding with the normal and tangent to the path at the point P. These components are called
normal acceleration and tangential acceleration respectively.

The total acceleration ‘a’ of a particle in curvilinear motion is a vector. This acceleration may arise
as a result of change in magnitude of velocity or change of direction of velocity or both. When we
resolve this total acceleration into normal and tangential components, we separate these two effects.
The change in speed is accounted for by the tangential acceleration alone, while change of direction
of motion is accounted for by the normal acceleration alone.

To obtain these components , the total change in velocity ∆𝑣 as the particle moves from P to P1
and resolve this into components ∆𝑣𝑛 and ∆𝑣𝑡 parallel to the normal and tangent at P respectively,
then the corresponding components of the average acceleration ∆𝑣 will be

∆𝑣𝑡 ∆𝑣𝑛
(𝑎𝑡 )𝑎𝑣𝑒 = (𝑎𝑛 )𝑎𝑣𝑒 =
∆𝑡 ∆𝑡

∆𝑠
We know that the angle between the normal at p and p1 is ∆𝜃 = 𝑤ℎ𝑒𝑟𝑒 𝜌 is the radius of
𝜌
curvature of the path at P.

∆𝑠
∆𝑣𝑛 = 𝑣∆𝜃 = 𝑣
𝜌

∆𝑣𝑡 = 𝑣̅1 − 𝑣̅ = ∆𝑣 =change in speed of the particle when moves from P to P1

 ∆𝑣𝑡 ∆𝑣 ∆𝑣𝑛 𝑣 ∆𝑠
(𝑎𝑡 )𝑎𝑣𝑒 = = and (𝑎𝑛 )𝑎𝑣𝑒 = = 𝜌 ∆𝑡
∆𝑡 ∆𝑡 ∆𝑡

∆𝑣 𝑑𝑣
Instantaneous tangential acceleration 𝑎𝑡 = lim =
∆𝑡→0 ∆𝑡 𝑑𝑡

𝑣 ∆𝑠 𝑣 𝑑𝑠 𝑣2
Instantaneous normal acceleration 𝑎𝑛 = lim = 𝜌 𝑑𝑡 = .and it is directed towards center of
∆𝑡→0 𝜌 ∆𝑡 𝜌
curvature of the path.

The change in speed is accounted for by the tangential acceleration alone, while change of direction
of motion is accounted for by the normal acceleration alone.

Differential equation of curvilinear motion:

According to Newton’s second law of motion, its states that under the action of a force F a particle
receives an acceleration a which is in the direction of the force and proportional to its magnitude
𝑤
𝐹𝑥 = 𝑥̈
𝑔

𝑤
𝐹𝑌 = 𝑦̈
𝑔

For curvilinear motion, tangential and normal acceleration and corresponding components of
resultant force, equations of motion can be represented in the following form:
𝑊 𝑑𝑣
Tangential force S = 𝑔 𝑑𝑡

𝑊 𝑣2
Normal force N = 𝑔 𝜌

Motion of a projectile:

Projectile:The freely projected particles in the air in the direction other than vertical are having
combined effect of a vertical and horizontal motion is called projectiles.

velocity of projection: The velocity with which the particle is projected is called as velocity of
projection.

angle of projection: The angle between the direction of projection and horizontal is called angle of
projection.

trajectory :The path traced by the projectile is called as its trajectory and it is a parabola.

Range : the horizontal distance through which the projectile travels in its flight is known as the
range of the projectile

Time of flight is the time during which the projectile is in motion.

Maximum height is the highest elevation reached by the projectile above the ground during its
motion.
Motion of the projectile: Equation of trajectory of projectile:

Let us consider the motion of a projectile fired from the origin of co-ordinates with an initial velocity
vo inclined to the horizontal by the angle α and having the projections 𝑥̇0 𝑎𝑛𝑑𝑦0̇ .

Neglecting air resistance on the projectile and assuming that only the vertical gravity force W is
acting, the equations of motion is
𝑤 𝑤
𝐹𝑥 = 𝑥̈ = 0, 𝐹𝑦 = 𝑦̈ = -w
𝑔 𝑔

Or 𝑥̈ = 0 𝑦̈ = -g

Integrating these equations once, we obtain

𝑥̇ =̇ 𝑐1 𝑦̇ = -gt + c2

Knowing that at the initial moment when t=o, 𝑥̇ = 𝑥̇ 0 and 𝑦̇ = 𝑦̇ 0

Then 𝑐1 = 𝑥̇ 0 𝑐2 = 𝑦̇ 0

And 𝑥̇ = 𝑥̇ 0 𝑦̇ = -gt + 𝑦̇ 0

These are the velocity time equation for projectile

A second integration gives

1
𝑥 = 𝑥̇ 0 𝑡 + 𝐷1 𝑦 = − 2 𝑔 𝑡 2 + 𝑦̇ 0 𝑡 + 𝐷2
Again, knowing that at the initial moment when t=o, we have x=y=o, we find for the constants of
integration

D1 =0 D2 =0
1
𝑥 = 𝑥̇ 0 𝑡 𝑦 = − 2 𝑔 𝑡 2 + 𝑦̇ 0 𝑡

These are the displacement - time equation for the projectile .For any value of t we can calculate
from then the co – ordinate x and y at the moving projectile and hence they completely define the
motion
𝑥
To find the equation of the path of the projectile, we eliminate t by using 𝑡 = 𝑥̇
0

1 𝑥2 𝑥
𝑦 = −2𝑔 + 𝑦̇ 0
𝑥̇ 0 𝑥̇ 0

𝒈𝒔𝒆𝒄𝟐 𝜶
Using 𝑥̇ 0 = 𝑣0 cos 𝛼 and 𝑦̇ 0 = 𝑣0 sin 𝛼 we have 𝒚 = 𝒙 𝐭𝐚𝐧 𝜶 − 𝒙𝟐
𝟐𝒗𝟐𝟎

This is the equation of a parabola having a vertical axis.

Expression for maximum height reached:

𝑑𝑦
At maximum height slope 𝑑𝑥 of the curve zero and we can obtain

𝑑𝑦 −𝑔 𝑦̇ 0
= 𝑥̇ 2 𝑥 + =0
𝑑𝑥 0 𝑥̇ 0

𝑥̇ 0 𝑦̇ 0 𝒚̇ 𝟐 𝑣02 𝑠𝑖𝑛2 𝛼
=> 𝑥1 = and 𝒚𝟏 = 𝟐𝒈𝟎 = = maximum height reached
𝑔 2𝑔

Expression for time of flight :

At maximum height vertical velocity is zero

𝑦̇ 0
𝑦̇ = -gt + 𝑦̇ 0 =0 => 𝑡 = 𝑔

Time of flight = time to reach max. height +time to reach from max. height to horizontal = t1+ t2 =
2𝑦̇0 2𝑣0 sin 𝛼
2t1=> 𝑡 = =
𝑔 𝑔

Expression for Range :

2𝑥̇0 𝑦̇ 0 2𝑣02 𝑠𝑖𝑛𝛼𝑐𝑜𝑠𝛼 𝑣02 sin 2𝛼

Range of body is equal to 𝑅 = 2𝑥1 = => 𝑅 = =
𝑔 𝑔 𝑔

It is seen that for a given initial velocity V0 the maximum range is obtained when α =450 and this
𝒗𝟐𝟎
maximum range is 𝑹𝒎𝒂𝒙 = 𝒈
D, Alembert’s principle

The equations of motion of a particle can be written in the form

𝐹𝑥 − 𝑚𝑥̈ = 0

𝐹𝑦 − 𝑚𝑦̈ = 0

In case of any rigid body that has curvilinear translations, all particle of the body have the same
motion and hence the same acceleration. If to each particle the inertia force is added. The resultant at
this inertia force balances the result of the active forces external to the body and we have ages a
system of forces in equilibrium. Internal forces between the various particles of the body always
occur in balanced pairs and need not be considered.

Work and energy in curvilinear motion:

Considering the equation of motion for a particle under curvilinear motion,

𝑊 𝑑𝑣
= 𝑆 ------------------------- ①
𝑔 𝑑𝑡

𝑊 𝑣2
= 𝑁 ------------------------- ②
𝑔 𝜌

Where, S and N are the projection of the resultant force F on the tangent and normal to the path at
any point P as shown in fig
𝑑𝑠 𝑊 𝑑𝑣 𝑑𝑠
Multiplying Eq. ① by = , we obtain 𝑣 = 𝑆 𝑑𝑡 ;
𝑑𝑡 𝑔 𝑑𝑡

This can be re written as,

𝑊 𝑉2
(𝑔 ) = 𝑆 𝑑𝑠 ------------------------- ③
2
 𝑊 𝑉2
In Eq. ③, the term, 𝑑 ( 𝑔 ) represents the change in the kinetic energy of the particle during the
2
time interval dt; and the term, 𝑆 𝑑𝑠 represents the corresponding increment of work done by the
tangential force ‘S’ acting through the displacement ‘ds’.

Integrating Eq. ③, between limits corresponding to any two parts 1 and 2 on its path, we obtain,

𝑤1 𝑣22 𝑤2 𝑣12 𝑆
− = ∫𝑆 2 𝑆 𝑑𝑠 ------------------------- ④
𝑔 2 𝑔 2 1

This is the equation of work and energy for the case of curvilinear motion of a particle in a
plane. It states that “the change in kinetic energy of the particle between any two positions on its
path is equal to the work of all forces acting upon it during the motion between these two
points.”

Instead of the tangential and normal component forces, if we consider the rectangular
component forces i.e. Fx and Fy of the force acting on the particle (As shown in fig. ) and also
considering the components of displacement ‘ds’ as ‘dx’ and ‘dy’, then we can write the work done
as

𝑆 𝑑𝑠 = 𝐹𝑥 𝑑𝑥 + 𝐹𝑦 𝑑𝑦 ------------------------- ⑤

Then Eq. ③, can be re written as

𝑊 𝑉2
𝑑 (𝑔 ) = 𝐹𝑥 𝑑𝑥 + 𝐹𝑦 𝑑𝑦 ------------------------- ⑥
2

Integrating Eq. ⑥, between limits corresponding to any two parts 1 and 2 on its path, we obtain,
𝑤1 𝑣22 𝑤2 𝑣12 𝑥 𝑦
− = ∫𝑥 2 𝐹𝑥 𝑑𝑥 + ∫𝑦 2 𝐹𝑦 𝑑𝑦 -------------------------⑦
𝑔 2 𝑔 2 1 1