Instantaneous velocity describes how fast an object is moving at different instants of time in a given time interval. It is also defined as average velocity for an infinitely small time interval.

Here lim is taking operation of taking limit with time tending towards 0 or infinitely small.

dx/dt is differential coefficient – Rate of change of position with respect to time at an instant.

The SI unit of work is the joule (J). ewton meter in termed and Joules and it is the unit of Work.

The SI unit of energy is joules (J), which is named in honour of James Prescott Joule.

Parallelogram Law of Vector Addition:

This law is also very similar to the triangle law of vector addition. Consider the two vectors again.
Now for using the parallelogram law, we represent both the vectors as adjacent sides of a parallelogram and then the diagonal emanating from the common point represents the sum or the resultant of the two vectors and the direction of the diagonal gives the direction of the resultant vector.

The resultant vector is shown by C. This is known as the parallelogram law of vector addition.

By using the orthogonal system of vector representation the sum of two vectors

a = a₁i^+a₂j^+a₃k^ and b = b₁i^+b₂j^+b₃k^ is given by adding the components of the three axes separately.

i.e. a + b = a_ii^+a₂j^+a₃k^+b₁i^+b₂j^+b₃k^

⇒a+b = (a₁+b₁)i^+(a₂+b₂)j^+(a₃+b₃)k^

Quantities Having the Same Dimensional Formula

1. Impulse and momentum.
2. Work, energy, torque, the moment of force, energy.
3. Angular momentum, Planck’s constant, rotational impulse.
4. Stress, pressure, modulus of elasticity, energy density.
5. Force constant, surface tension, surface energy.
6. Angular velocity, frequency, velocity gradient.
7. Gravitational potential, latent heat.
8. Thermal capacity, entropy, universal gas constant and Boltzmann’s constant.
9. Force, thrust.
10. Power, luminous flux.

Numerical analysis is concerned with the methods of finding the approximate values and the absolute errors in these calculations. The absolute error gives how large the error is, while the relative error gives how large the error is relative to the correct value. In numerical calculation, error may occur due to the following reasons.

• Round off error
• Truncation error

The velocity-time graph of a uniformly accelerated motion is a straight line graph inclining towards the time axis. If the object has positive constant acceleration, the graph slopes upward. In case the object has negative constant acceleration, the velocity-time graph will slope downward. 

The pattern of slope of the graph shows that object is moving with uniform acceleration. Calculation of Displacement and Distance covered by the moving object using velocity time graph: Let take two points, A and B at the slope of the graph. Draw a line from B to BD and another from point A to AE parallel to y-axis.

Dimension of Velocity is LT⁻¹ 
Dimension of wavelength λ is L 
Dimension of acceleration due to gravity g is LT⁻² 
Dimension of Density of water ρ is ML⁻³
Let v∝λ^ag^bρ^c
Using Dimensional method:
a+b−3c=1  ( Length ) 
c=0 ( Mass ) 
−2b=−1  ( Time ) 
b=0.5 
Hence, a=0.5
∴v∝ √gλ​
v²∝gλ

The kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. 

Consider a body of mass "m" starts moving from rest. After a time interval "t" its velocity becomes V.
If initial velocity of the body is Vi = 0 ,final velocity Vf = V and the displacement of body is "d". Then

Derivation for the equation of Kinetic Energy:
The relation connecting the initial velocity (u) and final velocity (v) of an object moving with a uniform acceleration a, and the displacement, S is 
                             v² - u² = 2aS
This gives  
                             S = ½a(v² - u²) 
We know F = ma. Thus using above equations, we can write the workdone by the force, F as
                            W = ma × ½a(v² - u²) 
                                  or
                             W =m( v 2 - u 2 ) ½

If object is starting from its stationary position, that is, u = 0, then
                            W = ½mv²
It is clear that the work done is equal to the change in the kinetic energy of an object.
If u = 0, the work done will be  W = ½mv²
Thus, the kinetic energy possessed by an object of mass, m and moving with a uniform velocity, v is E_k=  ½mv²

The coefficient of viscosity is a measure of resistance to flow of the fluid. ... Dynamic viscosity μ is used in Darcy's law to calculate the rate of fluid flow in porous media. The relationship between viscosity and flow rate defines the rheology of the fluid. The SI unit of viscosity coefficient (η) is Nm-2s or Nsm-2.  It is the ratio of the actual velocity to the theoretical velocity of a fluid jet.

Adiabatic Processes

• Adiabatic is a process in which there is no heat flow takes place between the system and the surroundings.
• These processes are sudden.
• The walls of the container should be adiabatic
• For an adiabatic process of an ideal gas

Addition of vectors satisfies two important properties.

• The commutative law, which states the order of addition doesn't matter: a+b=b+a. ...
• The associative law, which states that the sum of three vectors does not depend on which pair of vectors is added first: (a+b)+c=a+(b+c).
• Triangle law of vector addition states that when two vectors are represented as two sides of the triangle with the order of magnitude and direction, then the third side of the triangle represents the magnitude and direction of the resultant vector.