Linear Motion Details

Linear Motion Details

Introduction:

In the universe, matter and energy are the most fundamental things. Physics deals with the inter-relationship between matter and energy. In physics we study about the nature of matter and energy and we study the relationship between the two. Hence, physics is considered to be the most fundamental science.

Linear Motion Details

 Rest:

When the body does not change its position with respect to the surroundings, the body is said to be at rest.

Example:

A person, sitting inside a moving bus, is at rest with respect to the person sitting next to him as he is not changing his position.

Motion:

When the body changes its position with respect to its surrounding, the body is said to be in motion.

 Example:

Considering Sun as reference point, planets change its position, so, planets are in motion.

 

Velocity:

The rate of change of displacement is called velocity. Velocity is the speed of an object and a specification of its direction of motion. Speed describes only how fast an object is moving; whereas velocity gives both how fast and in what direction the object is moving. The velocity of a moving body is defined as its rate of displacement. The velocity may be uniform or variable. It is donated by. It is a vector quantity and its unit is ms^-1.

Uniform Velocity:

A body is said to move with uniform velocityif it has no acceleration. This implies that the body moves with a constant speed along a straight line path. This also means that the body moves with equal displacements in equal intervals of time, however small the time intervals

may be. “Uniform velocity” means zero acceleration, that is, constant speed in a straight line, The Rate of change of velocity is zero. The velocity of sound, the velocity of light etc. are the example of uniform velocity.

Variable Velocity:

If the velocity of body of a body is different at different time then that velocity is called variable velocity. Velocity in which the space described varies from instant to instant,

either increasing or decreasing; in the former case called variable velocity.

 

Instantaneous Velocity:

The velocity of an object at a particular instant or at a particular point of its path is called instantaneous velocity. In another word, the instantaneous velocity of an object is defined as the limiting value of the average velocity of the object in a small time interval around that instant, when the time interval approaches zero.

 

Speed:

The rate of change of distance or position of a particle is called speed. It is a scalar quantity. Speed is always positive. It’s unit is ms^-1.

Acceleration:

Acceleration is the rate of change of velocity as a function of time. It is vector. In calculus terms, accelerationis the second derivative of position with respect to time or, alternately, the first derivative of the velocity with respect to time.

If the change of velocity is, time variationthen the acceleration will be

 

Uniform Acceleration:

If an object’s speed (velocity) is increasing at a constant rate then we say it has uniform acceleration.

The rate of accelerations constant.

If a car speeds up then slows down then speeds up it doesn’t have uniform acceleration.

Displacement:

The minimum distance between two points is known as Displacement. The vector distance between two points is known as Displacement.It is denoted by r or s or d. It is a vector quantity. It’s unit is meter. Displacement, s = v t, where, v = velocity and t= time.

 

Relative Velocity:

Relative velocity is a measurement of velocity between two objects moving in different frames of reference. Relative velocity is an essential area of both classical and modern physics, since nearly all non-trivial problems in physics deal with the relative velocity of two or more particles. It is especially important in special relativity where there is no such thing as absolute motion, thus making all motion and therefore velocities relative. Since velocity is the change in position with respect to time, two velocities, v and w could be alternatively written as the derivative of the position with respect to time.

Distinction between Speed and Velocity:

Sl. No	Speed	Velocity
1.	The rate of change of distance or position of a particle is called speed.	The rate of change of displacement is called velocity.
2.	It is a scalar quantity.	It is a vector quantity.
3.	It is measured by speedometer.	It is measured by velatometer.
4.	Speed is always positive.	Velocity may be positive or negative.
5.	The addition, substraction of it is done by algebraic rules.	The addition, substraction of it cannot be done by algebraic rules.

Distinction between Velocity and Acceleration:

Sl. No	Velocity	Acceleration:
1.	The rate of change of displacement is called velocity.	Acceleration is the rate of change of velocity as a function of time.
2.	Velocity is donated by v.	Acceleration is donated by a or f.
3.	Unit of velocity is ms-1.	Unit of acceleration is ms-2
4.	Dimension of velocity = [LT -1].	Dimension of acceleration = [LT -2].

 

Linear Motion Details

(a) Derivation of equation, v_x = v_{xo +} a_xt

Let, a body moves with uniform acceleration a_x, Let, at time t = 0 then initial position x = 0 and Initial velocity v_x = v_xo, Again at time t = t then final position x = x  and final velocity

v_x = v_x

In physics, acceleration is the rate at which the velocity of a body changes with time.

⇒ dv_x = a_x dt

When, t = 0  then v_x = v_xo  and when, t = t  then v_x = v_x  in this range, integrate the above equation,

⇒ v_x – v_xo = a_x (t – o)

∴ v_x = v_xo + a_xt

 

(b) Derivation of equation, x= x_o + ½(v_xo+v_x)t

Let, a body moves along X -axis with uniform acceleration. Let, at initial time t = 0 then initial position

x = x₀  and Initial velocity v_x = v_xo  Again final at time t = t then final position x = x₀ and final velocity v_x = v_xo

So, after t second the average velocity of the particle

∴  x= x_o + ½(v_xo +v_x)t

 

(c) Derivation of equation, x= x_o +v_xot+½a_xt²

Let, a body moves along X -axis with uniform acceleration a_x. Let, at time t=0 then initial position x = x₀ and Initial velocity v_x = v_xo, Again at time t = t then final position x = x and final velocity v_x = v_x.

In physics, acceleration is the rate at which the velocity of a body changes with time.

⇒ dv_x = a_x dt

When t = 0  then v_x = v_xo and when  t = t then v_x = v_x in this range integrate the above equation we get,

⇒ v_x – v_xo = a_x (t- o)

∴ v_x = v_xo + a_xt … … … (1)

Since, the velocity of a moving body is defined as its rate of displacement so by the definition,

⇒ dx = v_xo dt + a_xt dt

When, t = 0  then x = x₀  and when   t= t  then x = x  in this range integrate the above equation we get,

⇒ x – x_o = v_xo (t-o) +½a_x (t² – 0)

⇒ x = x_o + v_xo t + ½a_x t² (Answer)

Again, ⇒ x – x_o = v_xo t + ½a_x t²

∴ S = v_xo t + ½ a_x t²      [ ∵ x – x_o = S]

Let, a body moves along X -axis with uniform acceleration  a_x  Let, at time t = 0 then initial position x = 0 and Initial velocity v_x = v_xo  Again at time t = t then final position x = x and final velocity  v_x = v_x.

In physics, acceleration is the rate at which the velocity of a body changes with time.

⇒ v_x dv_x = a_x dx

When  x = x_o then v_x = v_xo   and when, x = x then v_x = v_x  in this range, Integrate the above equation we get,

 

Laws of falling bodies:

During free fall, the bodies obey three laws. These laws are known as laws of falling bodies. In 1589 A.D. Galileo was the first to demonstrate and then formulate these equations.

 

1st law:

All the freely falling bodies traverse equal distance at equal interval of time.

 

2nd law:

The velocity of the freely falling bodies is proportional to the time taken to fall. If the velocity of the falling bodies is v and time taken to fall t then according to 2nd law, v ∝ t

 

3rd law:

The distance traversed by the freely falling bodies is proportional to the square of the time of fall. If the distance traversed by the falling bodies h and time taken to fall is t, then according to 3rd law h∝t²

Guinea and feather experiment:

Newton proved the first law of freely falling bodies by this experiment.

He removed air from a tall hollow glass tube which contained a guinea

and a feather. On inverting the tube he found that both the feather and

the guinea reach the bottom of the tube at the same time. It is proved

that, in vacuum all bodies travel equal distance in equal interval time.

In 1971, astronaut R. Scott released a hammer from one hand and a

feather from the other on the moon and observed that both fall together.

Derivation of the equations of motion by (v – t) graphical method

(1) Derivation of v = v_o+at

The velocity-Time graph of a body shown in (fig. X) the initial velocity of the body is v₀ at point A. From A to B the velocity changes at a uniform rate that means; there is a uniform acceleration a from A to B in time t its final velocity becomes v which is equal to BC in the graph (fig. X). The perpendicular is drawn from point C. AD parallel to OC and perpendicular BE from point B to OE is drawn. From (fig. X)

Initial velocity,

v_o = OA = DC…. …. …. … (1)

Final velocity,

v = BC ….. …. ….. … … .. (2)

But, BC = BD + DC … … ..(3)

So, v = BD+DC

Or, v = BD+ v_o … … … … (4)

We know, acceleration, a= slope of line AB from (fig. X)

But, AD = OC= t

Or, BD = at

Now we get from the equation (4)

v = at + v₀ ∴ v = v₀ + at

(2) Derivation of equation, s = v_o t+ ½at²

Let the body travel a distance s in time t. In (fig. X) the distance travelled by the body is given by the area of the space between AB and OC= area OABC

Distance travelled = Area OABC

s= Area of the rectangle OACD + Area of the triangle ABD

Or, s = OA× OC+½× AD× BD

Or, s = (v_o× t) + ½× t× at [∵ AD=t and BD = at]

∴ s = v_o t + ½ at² … … … (5)

From (fig. X) we have seen that, distance (s) travelled by the body in time

t= area of the trapezium OABC

Now, we eliminate tfrom equation (6)

Putting the value of tin equation (6)

 

Linear Motion Details

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Linear Motion Details

Linear Motion Details

Linear Motion Details

Linear Motion Details

Linear Motion Details

Linear Motion Details