MOTION

Motion 

If the position of an object changes with time and its surroundings, the object is said to be in motion. The nature of the motion of an object can be studied by plotting of distance-time graph.

Displacement 

 It is the shortest distance between the initial and final positions traveled by the object. 

• It is not necessary that the distance covered by body is path length and it is always positive.
• If the motion is along a fixed direction, the distance and displacement of a moving object have the same magnitude. 
  
• When the object moves along the straight line in the same fixed direction.
• Displacement can not be greater than the distance traveled by an object, because displacement is always less than or equal to the distance traveled, if they are the same direction, then their magnitude will be equal, however, if they have different directions then displacement will be less than the distance.
• It is a vector quantity. 
• S.I unit is m. 

Speed 

The distance covered by a body in a unit interval of time is called speed. It is a scalar quantity. 

speed = Distance / Time

S.I unit is m/s

Uniform and nonuniform speed

If a body covers equal distance in equal interval of time throughout its motion is said to be uniform speed and if If a body covers the unequal distance in equal interval of time throughout its motion is said to be nonuniform speed.

Average speed

The ratio of the total distance traveled by the body to the total time of the journey is called average speed.
Av. Speed = Total distance / Total time.

Velocity 

The distance traveled by a body in a specified direction is called velocity. It is a vector quantity. S.I unit is m/s. 
Velocity = Displacement / Time.

Uniform velocity and non-uniform velocity

If a body covers equal distance in equal interval of time throughout its motion along a particular direction, is said to be uniform velocity and if If a body covers the unequal distance in equal interval of time throughout its motion in a particular direction is said to be non-uniform velocity. 

Average velocity 

The ratio of the total displacement traveled by the body to the total time of the journey is called average velocity.

Av. velocity = Total displacement / Total time.

Instantaneous velocity 

 When a body moving with a variable velocity, the velocity of the body at any instant is called instantaneous velocity.

Odometer

It measures distance traveled by automobile.

Speedometer - 

Rest and motion relative terms

Rest and motion are relative terms. While sitting on a moving bus our distance from the walls, roof, and floor of the bus does not change. So with respect to the bus our position does not change. Therefore we are at rest with respect to the bus but our distance from the bus stand changes with time. So we are moving with respect to the bus stand. So motion and rest are relative terms.

Differences between speed and velocity.

• Speed is the rate of change of motion and velocity is the rate of change of motion in a particular direction.
• Speed is scalar quantity and velocity is a vector quantity.
• Speed cannot be zero or negative but velocity can be zero, positive or negative.

Circular motion is accelerated motion

A particle moving in a circular path changes its direction continuously. its velocity therefore not constant even if its speed is constant. so it is an example of accelerated motion.

Expression of circular motion (v) = 2πr/t   Where r= radius and t = time.

Uniform circular motion.

If a body moves in a circular path and covers equal distance in equal interval of time, motion is called uniform circular motion.

Two examples are: Revolution of earth and the revolution of the moon.

Scalar and vector quantity.

• A scalar is a quantity that has only magnitude or numerical value. Ex- Temperature and mass are scalar quantity. 
• A vector is a quantity that has both magnitude and direction. Ex-Velocity and acceleration is a vector quantity.

Acceleration 

The rate at which the velocity of an object changes, is called the acceleration of the object, so when the particle moving with a uniform velocity its acceleration will be zero.

 Acceleration (a) = Change in velocity/time.

S.I unit is m/s^2.

Negative acceleration 

An object which moves in the positive direction has a positive velocity and if the speed of the object decreases, it is called negative acceleration or deceleration or retardation.

Uniform acceleration and nonuniform acceleration

When the equal change in velocity takes place in an equal interval of time, the acceleration is called uniform acceleration.

When an unequal change in velocity takes place in an equal interval of time, the acceleration is called non-uniform acceleration.

Retardation ( deceleration)

If the velocity of a body decreases with time, the motion is said to be declaration or retardation.

Differences

Distance and displacement.

• Distance is the length of the path traversed by the object in a certain time but displacement is the shortest distance between the initial and final position of the object.
• Distance is a scalar quantity, it has only the magnitude but displacement is a vector quantity and has both the magnitude and direction.
• Distance is always positive but displacement can be positive, negative or zero.
• The distance can be more than or equal to the displacement but displacement can never be greater than the distance, it can be equal or less than distance.

Speed and velocity
Difference between speed and velocity - 

• The rate of motion of an object or the distance traveled by the object in unit time is called speed. The SI unit of speed is meter per second but the velocity is the rate of motion of an object or the distance traveled by the object in a particular direction in unit time is called velocity.
• Speed is a scalar quantity but velocity is a vector quantity.
• Speed is always positive but velocity can be positive, negative or zero.

Uniform and non-uniform motion.
Uniform and non-uniform motion- If a moving object covers equal distances in equal intervals of time, it is said to be in uniform motion. but if the object covers unequal distances in equal intervals of time, it is said to be in non-uniform motion. 

Derive graphically the first equation of motion:

V = u + at
S= Ut + 1/2 at²
v² - u² = 2as.

1. DERIVATION OF EQUATION FOR VELOCITY - TIME RELATION :

Consider the velocity-time graph of an object that moves under uniform acceleration as shown in Fig. From this graph:
The velocity changes at a uniform rate. 
In Fig. the initial velocity (u) is represented by OA, the final velocity(v)is represented by BC and the time interval t is represented by OC.
BD = BC – CD (the change in velocity in time interval t)
AD parallel to OC and OC = AD = t 
From the graph, we observe that:
 BC = BD + DC
= BD + OA
Substituting BC = v and OA = u, we get 
v = BD + u or BD = v – u
The acceleration of the object is given by
a = Change in velocity /time taken 
= BD = BD = BC - DC
   AD     OC        OC
a = BC - DC
          OC
Substituting OC = t, BC = v and DC = u, we get 
a =  v - u
          t
or v - u = at
v = u + at 

2. DERIVATION OF EQUATION FOR POSITION-TIME RELATION 

Let us consider that the object has traveled a distance s in time t under uniform acceleration a. In Fig. 8.8, the distance traveled by the object is obtained by the area enclosed within OABC under the velocity-time graph AB. Thus, the distance s traveled by the object is given by 
s = area OABC (which is a trapezium) 
= area of the rectangle OADC + area of the triangle ABD
= OA × OC + 1/2 (AD × BD)
s = u × t + 1/2x t×at   (We have v = u + at and at = v-u =BD, therefore BD=at)
s = ut + 1/2 at²

3. DERIVATION OF EQUATION FOR POSITION–VELOCITY RELATION

From the velocity-time graph shown in fig,the distance s travelled by the object in time t, moving under uniform acceleration a is given by the area enclosed within the trapezium OABC under the graph. 
s = area of the trapezium OABC =1/2 (OA + BC) ×OC 
Substituting OA = u, BC = v and OC = t, we get
s = 1/2(u+v) t 
We have v = u + at 
at= v - u 
t = v - u
        a
s = 1/2 (v+u)(v-u)
                   a
2as = (v+u)(v-u)= v² - u²
2as = v² - u²

DERIVATION MATHEMATICALLY
V = u + at
S= Ut + 1/2 at²
v² - u² = 2as.

1. DERIVATION OF v = u + at

Let the time be t, Initial velocity u and final velocity v of a moving body.
We know that acceleration is the rate of change in motion.
a = final velocity - initial velocity / time =  v - u /t
at = v - u

v = u + at ..... equation (i)

2. DERIVATION OF S= Ut + 1/2 at²

We know that Average velocity = Total distance / Total time 

= Initial velocity + final velocity
                  2
= u + v 
      2
Distance s = Av. velocity x time = (u + v)/2 x t
2s = (v + u) x t
2s = (u + at + u) x t   [ v = u + at ]
2s = 2ut + at²
s = ut + 1/2at²..... equation (ii)

3. DERIVATION OF v² - u² = 2as.

We know that Average velocity = Total distance / Total time 
= Initial velocity + final velocity
                  2
= u + v 
      2
Distance s = Av. velocity x time = (u + v)/2 x t
2s = (v + u) x t
2s = (v+ u) x (v - u)/a      [ v = u + at and t = v - u/a]
2s = v² - u² /a
2as = v² - u² ..... equation (iii)