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Kinetic energy

The work energy (WE) theorem

Work done by a variable force

Work energy theorem for a variable force

Potential energy

Conservation of total mechanical energy

The potential energy of a stretched spring

Plots of the potential energy and kinetic energy of a spring

Work done by combination of conservtive and non-conservative forces

Various forms of energy

Conservation of total energy


Collisions in one dimension

Inelastic collisions

Completely inelastic collision

Elastic collisions

Coefficient of restitution

Collisions in two dimensions




Work is done by a force on the body over a certain displacement.

The work done by the force is defined to be the product of component of the force in the direction of the displacement and the magnitude of this displacement. Thus

If we take the vectors F and s

W = F . s = F s cos θ, where θ is the angle between F and s.

No work is done if,

  1. the displacement is zero

  2. the force is zero.

  3. the force and displacement are mutually perpendicular.

For the block moving on a smooth horizontal table, the gravitational force m g does no work since it acts at right angles to the displacement.

Assuming the moon’s orbits around the earth is perfectly circular then the earth’s gravitational force does not do any work.

Work can be both positive and negative. If θ is between 0o and 90o, cos θ is positive. If θ is between 90o and 180o, cos θ is negative.

The frictional force opposes displacement and θ = 180o, hence the work done by friction is negative (cos 180o = –1).

Kinetic Energy

Energy of a body by virtue of its motion is called kinetic energy.

The Work Energy (WE) Theorem

The change in kinetic energy of a particle is equal to the work done on it by the net force.

Work and energy have the same dimensions, [ML2T–2].

The SI unit of both work and energy is Joule (J).


As per third equation of rectilinear motion,

where u and v are the initial and final speeds and s the distance traversed.

Multiplying both sides by m/2, we have

Where the last step is from Newton’s Second Law, i.e., F = m a.


where Ki and Kf are the initial and final kinetic energies of the object, respectively.

Work Done By a Variable Force

Consider a plot of x and F(x).

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We can divide the area into area elements of equal widths, Δx.

If the displacement Δx is small, we can take the force F(x) as approximately constant and the work done is then

Adding successive rectangular areas, we get the total work done as

W  limΔx  0Σxixf Fx Δx

If we take limit as displacements approach zero, then the sum approaches a definite value equal to the area under the curve or the integral,

 W = xixfFx dx

Work Energy Theorem for a Variable Force

The time rate of change of kinetic energy is

dKdt=ddt12mv2=m dvdt v=F v



Integrating from the initial position (xi ) to final position ( xf ), we have

KiKfdK =xixfFx dx 

Potential Energy

Potential energy is the stored energy of an object. It is the energy by virtue of an object's position or configuration relative to other objects.

Potential energy is associated with restoring force.

Examples of potential energy are gravitational potential energy, electric potential energy energy stored in a stretched spring, etc.

Mathematically, the potential energy V(x) is defined if the force F(x) can be written as

Fx = -dVdx     

 xixfFx dx=-ViVf dV = Vi- Vf 

The change in potential energy, for a conservative force, ΔV is equal to the negative of the work done by the force

Gravitational potential energy

where h is the height of the object from the surface of the earth.

Conservative force

Conservation of Total Mechanical Energy

The total mechanical energy of a system is conserved if the forces, doing work on it, are conservative.

Example – Consider a ball is dropped from height h.

TE at the top when the ball is at rest is PE = mgh

When the ball reaches the ground the PE becomes 0 and all the PE is converted to KE.

Just before hitting the ground,

12mv2 = mgh 

Example – Vertical circle

The potential energy of the system is taken to be zero at the lowest point A.

At A,

E =12m vA2


TA  mg =mv2r

At C,

    Tc = 0

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E = 12m vc2 + 2 mgr


mg =mvc2r

 vc =gr

 E = 52mgr

Comparing with the equation at A, we have

12m vA2=52 mgr 

 vA =5gr

At point B,

E = 12m vB2 + mgr

52 mgr mgr  =12m vB2 

 vB =3gr

The Potential Energy of a Stretched Spring

The spring force is conservative. The system consists of a block attached to a spring and resting on a smooth horizontal surface. The other end of the spring is attached to a rigid wall.

Spring force is given by Hooke’s law

The constant k is called the spring constant. Its unit is N m-1. The spring is said to be stiff if k is large and soft if k is small.

If the block is pulled outwards and the maximum extension is xm, the work done by the spring (restoring) force is

Ws= 0xmFx dx

 Ws= -kxm22

The work done by the external pulling force F is positive.

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The same is true when the spring is compressed with a displacement xc (< 0).

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If the block is moved from an initial displacement xi to a final displacement xf , the work done by the spring force Ws is

Ws= xixfFx dx


Thus the work done by the spring force depends only on the end points. Specifically, if the block is pulled from xi and allowed to return to xi; Ws = 0.

⇒ Hence, the spring force is a conservative force.

Plots of the Potential Energy and Kinetic Energy of a Spring

Plots of the potential energy V and kinetic energy K of a block attached to a spring obeying Hooke’s law, are parabolic. The two plots are complementary, one decreasing as the other increases. The total mechanical energy E = K + V remains constant.

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Work Done by Combination of Conservtive and Non-conservative Forces

If the two forces on the body consist of a conservative force Fc and a non-conservative force Fnc, the conservation of mechanical energy formula will be,

But Fc Δx = − ΔV

Hence, Δ(K + V) = Fnc Δx

Where, Wnc is the total work done by the non-conservative forces over the path.

Various forms of energy

Energy exists in various forms which can transform into one another.

  1. Mechanical Energy – KE and PE

  2. Heat Energy

  3. Electrical Energy

  4. Light Energy

  5. Chemical Energy

  6. Nuclear Energy

Conservation of Total Energy

The total energy of the universe is constant. If one part of the universe loses energy, another part must gain an equal amount of energy.

The principle of conservation of energy cannot be proved. However, no violation of this principle has been observed.


Power is defined as the time rate at which work is done or energy is transferred.

The average power of a force is defined as the ratio of the work, W, to the total time t taken




Work dW done by a force F for a displacement dr is, dW = F.dr. The instantaneous power can therefore be expressed as


Power is a scalar quantity. Its dimensions are [ML2T–3].

The SI unit is called a watt (W).


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Consider two masses m1 and m2. The particle m1 is moving with speed u1 and m2 is at rest.

The mass m1 collides with the stationary mass m2 and m1 and m2 move with velocities v1 and v2 respectively, after the collision. v1 and v2 make angle θ1 and θ2 respectively with the original direction of u1.

In all collisions the total linear momentum is conserved, that is, the initial momentum of the system is equal to the final momentum of the system.

Δp1 = F12 Δt

Δp2 = F21 Δt

where F12 is the force exerted on the first particle by the second particle and F21 is the force exerted on the second particle by the first particle.

Now from Newton’s Third Law, F12 = − F21. This implies

Δp1 + Δp2 = 0

The total kinetic energy of the system is not necessarily conserved. The impact and deformation during collision may generate heat and sound. Part of the initial kinetic energy is transformed into other forms of energy.

Collisions in one dimension

θ1 = θ2 = 0

Inelastic collisions

If the kinetic energy is not conserved, the collision is called inelastic collision. That is the KE before and after collision is not the same.

Completely Inelastic Collision

A collision in which the two particles move together after the collision is called a completely inelastic collision.

By conservation of momentum,

m1u1 = (m1 + m2)v [v is same for both]

 v= m1m1+m2 u1

The loss in kinetic energy on collision is

ΔK =12m1u12-12m1+m2v2

Putting the value of v,

=12m1u12-12m1+m2 m12u12(m1+m2)2

=12m1u12(1- m1m1+m2)

=12 m1m2m1+m2u12

Elastic Collisions

If the kinetic energy is conserved, the collision is called elastic collision. That is the KE before and after collision is same.

By conservation of momentum,

m1u1 = m1v1 + m2v2

⇒ m1 (u1 - v1) = m2v2 -(1)

By conservation of kinetic energy,

m1u12=m1v12+ m2v22

   m1u12-v12= m2v22    -(2)

Dividing relation (2) by (1)

u1 + v1 = v2

Putting the value of v2 in (1)




Case I:

If the two masses are equal v1= 0

v2 = u1

The first mass comes to rest and pushes off the second mass with its initial speed on collision.

Case II: If one mass dominates, e.g. m2 > > m1

v1 ~ − u1 and v2 ~ 0

The heavier mass is undisturbed while the lighter mass reverses its velocity.

Coefficient of restitution e

e=v2- v1u2- u1

The coefficient is related to energy by

e=KE after the collisionKE before the collision

Range of values for e

e is usually a positive, real number between 0 and 1.

e = 0: This is a perfectly inelastic collision. The objects do not move apart after the collision, but instead they coalesce. Kinetic energy is converted to heat or work done in deforming the objects.

0 < e < 1: This is a real-world inelastic collision, in which some kinetic energy is dissipated.

e = 1: This is a perfectly elastic collision, in which no kinetic energy is dissipated, and the objects rebound from one another with the same relative speed with which they approached.

e < 0: A collision in which the separation velocity of the objects has the same direction (sign) as the closing velocity, implying the objects passed through one another without fully engaging. This may also be thought of as an incomplete transfer of momentum. An example of this might be a small, dense object passing through a large, less dense one – e.g., a bullet passing through a target, or a motorcycle passing through a motor home or a wave tearing through a dam.

Collisions in Two Dimensions

Conserving the momentum in x - direction,

m1u1 = m1v1 cos θ1 + m2v2 cos θ2

Conserving the momentum in y - direction,

0 = m1v1 sin θ1 − m2v2 sin θ2

If the collision is elastic then,

m1u12=m1v12+ m2v22

Now we have four unknowns (v1, v2, θ1, θ2) but only 3 equations. If one of the variables is known, we can solve for other 3 variables.