## REST, MOTION AND REFERENCE FRAME

Kinematics is the branch of physics that deals with motion of bodies without inquiring its cause (the concept of forces).

A Particle is defined as matter of infinitesimally small size. Thus a particle has only a definite position but no dimension.

A certain amount of matter limited in all directions with finite size, shape occupying some definite space, is called a Body.

• Rest

A body is said to be at rest when it doesn’t changes its position with time. Example- A blackboard sticking to a wall for some time is said to be at rest when it doesn’t move from its position.

• Motion

A body is said to be in motion if the position of the body continuously changes with time. Example- Movement of a car w.r.t. an observer at rest.

• Frame of Reference :

To locate the position of a body relative to the reference body a system of coordinates fixed on the reference body is constructed. This is known as reference frame.

If two cars A and B move side by side in same direction with same speed, it would appear to the passengers of the cars that they are mutually at rest. Obviously, B is at rest relative to A. The reverse is also true.

Absolute rest or absolute motion is undefined. Motions are relative.

*NOTE: Description of state of motion of a particle requires a set of axis, w.r.t to which the state must be specified, otherwise it won’t make any sense. The inherent meaning of this statement is that we need a reference frame to determine whether a body is at rest or in motion.

### DISTANCE AND DISPLACEMENT

What is Distance?

Let a particle has a starting point at ‘A=2m’ and later it comes to final point ‘B=4m’ after turning round through ‘C=6m’.

The distance travelled here = AC + BC = 4 + 2 = 6m

“Distance is the total path length of the travelled by the particle in a given time interval.”

• Path length of a body is a positive scalar quantity which doesn’t decrease with time and can never be zero for any moving body.
• Magnitude of distance is greater than r equal to the magnitude of displacement.

What is Displacement?

It is the vector joining the intial and final position of an object during a time interval. The change in position of moving object is known as displacement.

• For a straight line motion, if a particle goes from A to B then,
• s = displacement = AB, the vector has only x component.

Hence its equal to the difference in X coordinates

S= Xb – Xa = ∆X

• If a particle goes from A to B along a curve in some time duration and if O is the origin then,

OA = initial position vector =r i

OB= final position vector =r f

AB= displacement of vector = OB – OA

s = rf ri

|S|= (|r f2| – |r i2|) 1/2

Distance vs Displacement

### AVERAGE AND INSTANTANEOUS VELOCITY/SPEED

• AVERAGE SPEED: It tells us how fast a particle moves in a particular interval.

It is a scalar quantity and is defined over an interval as Average Speed = Total Distance Covered/ Total Time Interval

Average speed has unit (m s-1) and the magnitude of average speed is always greater than or equal to magnitude of average velocity.

• AVERAGE VELOCITY: The average velocity of a moving particle over a certain time interval is defined as the displacement divided by the time duration.

The average velocity is the vector in the direction of displacement. It depends only on the net displacement and time interval and not on the journey.

INSTANTANEOUS VELOCITY: The velocity at an instant is defined as the limit of the average velocity as the time interval ∆t becomes infinitesimally small. In other words ,

• Velocity at any instant is equal to the slope of tangent of displacement time graph.
• It is the average velocity for infinitely small time interval.
• Thus instantaneous velocity = tanθ =  ${{\frac {dx} {dt}}}$
• Magnitude of velocity in rectilinear motion at any time gives us speed of the particle.
• The SI unit of Velocity is m/s.

*Note that for uniform motion, velocity is the same as the average velocity at all instants. Instantaneous speed or simply speed is the magnitude of velocity. Average speed over a finite interval of time is greater or equal to the magnitude of the average velocity, instantaneous speed at an instant is equal to the magnitude of the instantaneous velocity at that instant.

### AVERAGE AND INSTANTANEOUS ACCELERATION

Average Acceleration: The rate of change of velocity for a moving particle with respect to time is known as average acceleration.

Instantaneous Acceleration:  The rate of change of velocity for a moving particle at any particular instant is known as instantaneous acceleration. Mathematically its expressed as ,

• It is the average acceleration for infinitely small time interval.
• Acceleration at any instant is equal to the slope of tangent of velocity time graph.
• Thus instantaneous acceleration = tanθ = ${\frac {dv} {dt}}$
• The SI unit of acceleration is m/s2.

If v = f(x) then , a = ${\frac {dv} {dt}}$ = ${\frac {dv} {dx}}{\frac {dx} {dt}}$ = $v{\frac {dv} {dx}}$

*When displacement is given and we need to calculate a, then we use

a = ${\frac {dv} {dt}}$ = ${\frac {{d}^{2}x} {{dt}^{2}}}$

### KINEMATICAL EQUATION FOR UNIFORMLY ACCELERATED MOTION

For uniformly accelerated motion, we can derive some simple equations that relate displacement (x), time taken (t), initial velocity (v0), final velocity (v) and acceleration (a).

Relation between final and initial velocities v and v0 of an object moving with uniform acceleration a can be derived from the relation:

a = ${\frac {dv} {dt}}$

Integrating both sides we get:

v = v0 + at

*This is called first equation of motion

• The area under the curve represents the displacement over a given time interval.

This 2nd equation of motion is graphically represented in Fig. 3.12.

The area under this curve is: Area between instants 0 and t = Area of triangle ABC + Area of rectangle OACD

X= ½(v-v0) t + v0t

But v-v0 = at Thus

x= v0t + ½* (a)* (t2)

For constant acceleration only

s= v0t + ½* (a)* (t2)

vavg = s/t = (v0t + ½* (a)* (t2))/t

vavg = u + ½ at = ½(2u+at) = ½(u + u + at)

vavg= ½(u+v)

Where u = initial velocity

And v= final velocity

x= vt = ½*(v + v0)*(v-v0) = (v2-v02) / 2a

v2-v20=2ax

This is 3rd equation of motion.
All the above sets of equations are obtained by assuming that at t= 0 the position of the particle is zero. In case there is an x0 displacement then the equations of motions are given by:

v = v0 + at

x =  x0 + v0t + ½* (a)* (t2)

v2-v20  =2a(x-x0)

### FREELY FALLING BODIES

When a body is dropped from some height which is much less than the radius of the earth, it falls freely under gravity with constant acceleration g = 9.8 m/s2 provided the air resistance is negligible.

The same set of kinematical equations are replaced with a = g and direction of y axis is chosen conveniently.

• Case-1: When   y axis is chosen positive for vertically downward motion then we take “g” as positive since the direction of g is always downwards towards earth.

v = v0 + gt

h =   v0t + ½* (g)* (t2)

v2-v20= 2g(h)

where h is the vertical displacement

Case – 2: When +ve  y axis is taken +ve for vertically upward motion  then the equations of motions are :

The equations of motions are:

v = v0 – gt

h =   v0t – ½* (g)* (t2)

v2-v20= -2g (h)

Note: Choosing of sign conventions is necessary. All the vector quantities should be assigned the same convention and the convention should be kept fixed throughout the problem. For example for case – 1 the displacement, velocity and g has a direction downward n we took downward as positive and rewrote the equations.

Calculating the Displacement during nth second

Displacement of n second- Displacement of (n-1)s

= un + ½(a)(n2) – { u(n-1) + ½(a)(n-1)2}

=u(n-n+1) + ½(a){n2 –( n-1)2}

sn= u + ½(a)(2n-1)

Calculation of stopping distance:

When brakes are applied to a moving vehicle, the distance it travels before stopping is called stopping distance. It is an important factor for road safety and depends on the initial velocity (v0) and the braking capacity, or deceleration, –a is caused by the braking.

Let the distance travelled by the vehicle before it stops be ds. Then, using equation of motion v2 = v02 + 2 ax, and noting that v = 0, we have the stopping distance

ds = -v2 / 2a

### POINTS TO REMEMBER WHILE PLOTTING GRAPHS

The theory of graphs can be generalised and summarised in following six points. :

• A linear equation represents straight line e.g. y = 4x. y= kx represents a straight line passing through origin in x- y graph.
• x= k/y represents a rectangular hyperbola in x-y graph.
• A quadratic equation represents a parabola in x- y graph .
• If  z= dy/dx , then value of z can be obtained by the slope of the graph at that point.
• If z = xy , then the value of z between x1 and x2 between y1 and y2  can be obtained by the area of graph between x1 and x2 and y1 and y2 .

Important points

• Slopes of v-t or s-t graph can never be infinite at any point, because infinite slope of v-t graph means infinite acceleration. Similarly, infinite slope of s-t graph means infinite velocity. Hence these graphs are not possible.
• At one time only two values of velocity or displacement are not possible.
• Different values of displacements in s-t graph corresponding to given v-t graph can be calculated just by calculating the area under v-t graphs.

### MOTION WITH NON-UNIFORM ACCELERATION

• Acceleration depends on time t

Steps to find v (t) from a (t) By definition we have

a = ${\frac {dv} {dt}}$

Now integrating both sides,

Where v0 = Initial velocity at time t=0

• Steps to find x(t) from v(t)

To get x (t), we put v (t) = dx/dt

dx= v (t)dt

Integrating both sides,

Where x (0) = Position at t= 0

### Relative Velocity

The word ‘relative’ means in relation or in proportion to something else.

Relative motion is the motion as observed from or referred to some system constituting a frame of reference.

The relative velocity of A with respect to B (written as ūAB ) means the velocity of A as seen from B . its represented as :

ūAB = ūA– ūB

Similarly, relative acceleration of A with respect to B is

āAB = āA– āB

For the further understanding of concepts related to equations of motion, graphical representation of motion in a straight line, relative velocity and freely falling bodies please refer to the PDF above in which these concepts are discussed in detail.

Categories: General