IIT JEE / CET – Main / Advanced Physics 2013 & 2014 Mechanics..Motion in one dimension

M O T I O N   I N  O N E   D I M E N S I O N

Maglev

Reference frame / Frame of reference : Defined as a set of axes, taken for practical purposes at being rest, which enables the position of a point ( or body ) , in space to be defined at any instant of time.To define the same,usually a set of four coordinate axes are taken,of which three are spatial and the fourth one is time. Basically of two types

  • Inertial frame of reference : Here , the frame of referance is either at rest or moving with constant velocity.
  • Non-inertial frame of reference : In this case, the frame of referance has some acceleration.

Reference frame in Physics : Here we look at some key points related to the utility and significance of the frames of reference in Physics.

Newton’s first law can be verified,only if the body considered is not acted on by any external forces.In such a case,what reference frame should we use ? May be , the reference frame should be fixed…… but fixed with respect to what ? . This is an important question because everything in universe is in relative motion and hence a special reference frame has to be thought of . Suppose , Newton’s first law is found to be true in a reference frame denoted F. This means that ,the above fact should be true in any other frame F′, which is mutually un -accelerated relative to F . It follows that. if a test particle has constant velocity in F , then it should have zero acceleration in F and also in F′ .

Therefore, an inertial reference frame is one in which the first law of Newton is true.The above points infer that, an exact inertial frame is not possible in nature . Now, the most practical reference frame available is EARTH . This is due to the fact that, when considering an earth bound body,the orbital acceleration of the earth is insignificant, and the effect of earth’s motion is only a small correction.But , when considering an orbiting satellite, a geo-centric frame (  with it’s origin at the centre of mass of earth and with no rotation relative to stars ) is needed. Also, it follows that, if we deal with planets, a helio-centric frame ( with it’s origin at the centre of mass of the solar system and no rotation relative to distant stars ) is required.

For taking observations, a Cartesian coordinate system is considered along with the frame of reference.This includes the three position coordinates  x,y and z and time. Now, at any particular time, one,two or all the three position coordinates may change.  i.e, there are three  dimensions of motion  :

  1. Motion in one dimension – when only one of the coordinates specifying a particle’s motion changes with time. Eg. motion along a straight path.
  2. Motion in two dimensions – here, any two coordinates specifying a particle’s motion changes with time. Eg.Projectile motion.
  3. Motion in three dimensions – When all the three co-ordinates specifying a particle’s motion changes with time Eg . brownian motion ( gas molecules )

Motion in physics can be breifly classified as  a) Translatory motion b)  Oscillatory motion and  c) Rotatory motion. All other motions are   considered as a combination of them.

  • Translatory motion : A body is said to undergo translatory motion , when it moves in such a way that the linear distance covered by each particle of the body is the same during the motion.Note that the body will not change it’s oreintation. If the motion is in a straight path, then it’s rectilinear and, if it’s along a curve, then it’s curvilinear.

Translatory motion
  • Oscillatory motion : The kind of movement when an object repeats it’s motion about a fixed point.The boundaries are called extreme points.
oscillatory motion
  • Rotatory motion : In this kind of motion, every particle of the body undergoes the same angular displacement about a particular axis of rotation.
Rotatory motion

Displacement : Could be defined as a specified distance in a specified direction and is the vector equivalent of the scalar distance . It’s a vector drawn from the initial position to the final position whose magnitude is equal to the shortest distance between the initial and final positions. [ note – distance refers to the actual path travelled by a particle during its motion in a given interval of time ].

Displacement, |s| = ∫|v|dt = ∫ ( νxiˆ+ νyjˆ + νzkˆ ) dt

Points to note :

  • Displacement is a vector quantity while distance is a scalar.
  • Displacement can be +ve , zero or -ve but distance is never -ve.
  • Displacement of a body for a particular motion is unique, eventhough it can have many distances.
  • Displacement = distance in uniform motion.
  • Displacement can decrease with time, but not the distance.
  • Displacement of a body ( magnitude ) , will never be greater than it’s distance.
  • Displacement is -ve if a body moves away from it’s initial position after reaching the same ,after covering a particular path.

Speed : The ratio of a distance covered by a body to the time taken or the time rate at which the distance is being travelled by a particle.

Unit  :  ms−¹ ( SI )   cms−¹  ( CGS )     Dimensions  :  [  M° L  T −¹ ]

  • Instantaneous speed – Usually , when one refers to speed, it is the instantaneous speed which is being considered.It simply refers to the speed at any particular instant of time.  Speed  =  Distance / Time.  Speed is scalar and is never negative.
  • Variable speed –   A particle covering equal distances in unequal intervals of time or unequal distances in equal intervals of time is said to undergo variable motion.
  • Uniform speed – A particle covering equal distances in equal intervals of time, however small these intervals may be , is said to undergo uniform speed.
  • Average speed – The ratio of the total distance travelled by a particle to the total time taken in which the distance is covered.Suppose a particle covers distances S1 S2 and S3 with velocities ν1 ν2 and ν3 , in time intervals of t1 t2 and t3 respectively. Then —

t1 = S1/ν1    ,     t2 = S2/ν2    and     t3 = S3/ν3 ,  average velocity upto S2  =

ν average  =  S1 + S2 / t1 + t2  =  ν1t1  + ν2t2 / t1 + t2             …………..   ( 1 )

= S1 + S2/  S1/ν 1 +  S2/ν 2                                                     ………….   ( 2 )

  1. When t1  =  t2  = t , then average velocity = v1 + v2 / 2    i.e, average speed is equal to the arithmetical mean of individual speeds.
  2. When S1 = S2  = S , then average speed =  2v1 v2 / v1 + v2  i.e, average speed is equal to the harmonic mean of individual speeds.

Uniform motion : A particle covering equal displacements in equal intervals of time , however small these intervals may be , is said to undergo  uniform motion.   Properties of uniform motion are —

  • Displacement is equal to the distance covered.
  • No net force is required for a body to undergo uniform motion.
  • The velocity is independant of the choice of origin.
  • The velocity is independant of the time interval.
  • It has average and instantaneous velocities.

Velocity : The rate of displacement of a body or  the speed of a body in a specified direction. As in the case of the def. of speed, velocity in most cases refers to instantaneous velocity which is defined as – Time rate of change of position ( x ) or displacement ( s ) at any instant of time ( t ).  Denoted by ν      = Displacement / time     =  dx / dt   or   ( ds/dt )  –  the rate of increase of the total distance travelled.  Velocity is a vector quantity and can either be +ve , zero or -ve .For a particle moving in three dimensional space,

|ν | =  d|r| / dt    =   d/dt [ xiˆ +  yjˆ +  zkˆ ]

=>  |ν|  =  vxiˆ  +  vyjˆ  + vzkˆ

[  Note  –  the direction of instantaneous velocity of a particle is always tangent to the particle’s path at the particle’s position ]

[  note – dimensions of velocity are same as that of speed ]

  • Uniform velocity : If a particle covers equal displacements in equal intervals of time , however small these intervals may be , then , it is said to undergo uniform  velocity. Here, the body continuosly moves in the same direction with constant speed.
  • Variable velocity : If a particle covers equal displacements in unequal intervals of time or  unequal displacements in equal intervals of time, it is said to have variable velocity.
  • Average velocity : defined as the ratio of the total displacement to the total time interval in which the displacement occurs. Note that , here we are not concerned about the nature of motion between the initial and final positions of the body. It could be also defined as the uniform velocity of a body in motion , where the body covers the same displacement in a given time as it would , with its actual velocity in the same time.

v av  = s/t  . If, for any time t 1 , the position vector of a particle is | r 1 | and at time t 2 , it’s | r 2 | ,  then

| v | av  =  | r 2 | –  |r 1 | / t 2 – t 1

average velocity

Acceleration : Rate of increase of speed or velocity.   Denoted by ‘ a ‘.

Units  –   m s-²  ( SI )  cms-² ( CGS )  Dimensions  –   M°  L  T-²

  • Instantaneous acceleration : The time rate of change of velocity at any instant of time. [ when we say acceleration, it usually refers to the instantaneous one ]. Could be defined as the limit of average acceleration at the time interval Δ t at which it becomes infinitesimally small.
    instantaneous acceleration

=> ax  = dvx/dt ,  ay = dvy/dt   and az = dvz/dt

Magnitude of a = √ ax² +  ay² + az²

  • a can be also expressed as –   v.dv/dx     a = d² x/ d t²
  • acceleration can also be defined as second time derivative of position.
  • acceleration is in the direction of increasing velocity.
  • acceleration is a vector quantity which can be +ve , zero or -ve . Negative acceleration is also called deceleration or retardation.
  • Average acceleration :  The ratio of the total change in velocity to the total time taken in which this velocity change takes place .

=  Total change in velocity / total time taken  =  Δv /Δ t

Note that the total change in velocity pertains only to the change in velocity between the initial and final positions.  Suppose , at an instant  t 1, the velocity of a particle is v1 and at instant t 2 , it’s v 2 , then the average acceleration of the particle is given by  –

  • Uniform / Constant acceleration : A particle is said to be moving in uniform acceleration if , equal changes in velocity takes place in equal intervals of time , however small these intervals may be. equal time periods. An example of an object having uniform acceleration would be a ball rolling down an incline. The object picks up velocity as it goes down the incline with equal changes in time.                                                                               

[ Note : Average velocity for an uniformely accelerated body is  –  Initial velocity + finalvelocity / 2  =  u + v / 2 ]

  • Variable acceleration  : In this case , the velocity of a particle changes equally in unequal intervals of time or velocity changes unequally in equal intervals of time. 
  • Angular acceleration :     The angular acceleration α is defined as the time derivative of the angular velocity ω.
  • Gravitational acceleration : it’s the acceleration on an object caused by gravity. In  vacuum, all small bodies accelerate in a gravitational feild at the same rate relative to the center of mass , irrespective of the mass or composition of the body. On the surface of the Earth, all objects fall with an acceleration between 9.78 and 9.82 m/s2 depending on the latitude, with a standard value of exactly 9.80665 m/s2 Naturally , objects with low densities do not accelerate as rapidly due to buoyancy and air resistance.
  • Acceleration in a plane curve : Consider an object accelerating along a curve in a plane. It’s acceleration will have both tangential as well as radial components and can be expressed as –    

where , ut – unit vector tangent to the path, pointing in the direction of motion at any particular instant of time , un – unit normal vector of the particle’s path [ pointing inward here ]  and R – radius of curvature.

Equations  of   Uniformely Accelerated Linear Motion :  Refers to equations describing the motion of bodies moving linearily [ straight line ] with uniform / constant acceleration. A commom method to remember them is the ‘ S U V A T ‘ mnemonic … i.e, S for displacement, U  for initial velocity, V for final velocity, A for acceleration and T  for the time taken..(  these are the constituents of the equations ).

The body is considered between two instants in time: one initial point and one current (or final) point. Problems may deal with more than two instants, and several applications of the equations are then required. If a is constant, a differential , a dt, may be integrated over an interval from 0 to Δtt = tti), to obtain a linear relationship for velocity. Integration of the velocity yields a quadratic relationship for position at the end of the interval.

v = v_i + a \Delta t \,
s = s_i + v_i\Delta t + \tfrac{1}{2} a(\Delta t)^2 \,
s = s_i + \tfrac{1}{2} (v + v_i)\Delta t \,
v^2 = v_i^2 + 2a(s - s_i) \,

where  vi is the initial velocity of the body, si is the initial position of the body, s is the final position , a is the acceleration , v is the final velocity and Δt is the time taken by the body to reach its final position. Note : in the above equations, each contain four of the five variables and hence by knowing any three , the other two can be found out.

The same for uniform circular motion would be –

\omega=\omega_0+\alpha t\,
\phi=\tfrac12(\omega_0+\omega)t
\phi=\omega_0t+\tfrac12\alpha t^2\,
\omega^2=\omega_0^2+2\alpha\phi\,
\phi=\omega t-\tfrac12\alpha t^2\,

where ω – final angular velocity, ω0 – initial angular velocity , Φ – angular displacement  α – angular acceleration and t  – time taken .

A more classic version is       —

\begin{alignat}{3} & v   && = u+at               \qquad & \text{(1)} \\ & s   && = \tfrac12(u+v)t     \qquad & \text{(2)} \\ & s   && = ut + \tfrac12 at^2 \qquad & \text{(3)} \\ & s   && = vt - \tfrac12 at^2 \qquad & \text{(4)} \\ & v^2 && = u^2 + 2as          \qquad & \text{(5)} \\ & a   && = \frac{v-u}{t}      \qquad & \text{(6)} \\ \end{alignat}

Note    ——–

  • by substituting ( 1 ) into ( 2 ) , we can get (3) , (4) and (5) .
  • rearranging (1) gives (6).
  • s  = displacement , u = initial velocity , v = final velocity , a = constant / uniform acceleration and t = time taken.
  • all the above equations are applicable only in non – relativistic velocities ( far less than the speed of light ).

Derivations    –

1 .   Velocity – time relation :      a =  v – u / t – 0        =>   v  = u + at

2 . Displacement – time relation :

 \mathrm{ average\ velocity } = \frac{s}{t}

Hence:

 \begin{matrix} \frac{1}{2} \end{matrix} (u + v) = \frac{s}{t}
s = \begin{matrix} \frac{1}{2} \end{matrix} (u + v)t

Using equation 1 to substitute u in equation 2 gives:

s = vt - \begin{matrix} \frac{1}{2} \end{matrix} at^2

3 .  Displacement – velocity relation :

t = \frac{v - u}{a}

Using equation 2, substitute t with above:

s = \begin{matrix} \frac{1}{2} \end{matrix} (u + v) ( \frac{v - u}{a} )
2as = (u + v)(v - u) \,
2as = v^2 - u^2 \,
v^2 = u^2 + 2as \,
( ………. to continue )

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