Motion In One Dimension |

 Introduction

If a body is in uniform motion (velocity = constant or body has acceleration and velocity acting at 0° or 180°, then path followed is straight line and motion is rectilinear motion.

Uniform motion

$\to$ Velocity is constant and acceleration is zero.

$\to$ Body covers equal displacement in equal time interval.

$\to$ Equation for uniform motion s = ut .

$\to$ time-displacement graph is straight line .

$\to$ Time velocity graph is straight line parallel to time axis

$\to$ Time acceleration graph is straight line coincide it to time axis.

Variable motion at constant acceleration

$\to$ Velocity changes at constant rate and acceleration is constant.

$\to$ Body may cover equal distances in equal time intervals but never covers equal displacements in equal time interval (a and v at 180°)

Distance and Displacement

$\to$ The length of actual path followed by a moving body is called distance. It is a scalar quantity.

$\to$ The shortest distance between initial and final position with direction for a moving body is called displacement . It is a vector quantity.

$\to$ Displacement covered by a moving body in certain time interval may be zero, negative or positive but distance never be zero or negative.

$\to$ Distance ≥ Displacement

$\to \dfrac{\text{Distance}}{\text{Displacement}} ≥ 1$

$\to$ For a moving body, the distance travelled always increases with time displacement may increase or decrease with time.

Speed and Velocity

Speed
$\to$ The time rate of change in distance for a moving body is called speed. It is a scalar quantity.
$\to$ Speed =$\dfrac{\text{distance}}{\text{time taken}}$
$\to$ The speed of a body can be zero or positive but never negative.
$\to$ A body is said to be moving with uniform speed if it covers equal distances in equal intervals of time.
$\to$ A body is said to be moving with a variable speed if it covers equal distances in unequal intervals of time or unequal distances in equal intervals of time.
$\to$ Average speed = $\dfrac{\text{distance covered in given time}}{\text{time taken}}$
$V_{av} = \dfrac{(x-x_0)}{(t-t_0)} = \dfrac{∆x}{∆t}$
Instantaneous speed is limiting value of average speed as time interval tends to zero.
$V_{av} = \displaystyle{\lim_{∆t \to 0}} \dfrac{∆x}{∆t} = \dfrac{dx}{dt}$
It is defined for a point of time .

Velocity

$\to$ The time rate of change in displacement tor a moving body is called velocity . It is a vector quantity .

$\to$ Velocity=$\dfrac{\text{displacement}}{\text{time taken}}$

$\to$ Average velocity = $\dfrac{\text{displacement covered in given time}}{\text{time taken}}$

$V_{av} = \dfrac{(s-s_0)}{(t-t_0)} = \dfrac{∆s}{∆t}$

It is defined for period of time.

Instantaneous speed is limiting value of average speed as time interval tends to zero.

$V= \displaystyle{\lim_{∆t \to 0}} \dfrac{∆s}{∆t} = \dfrac{ds}{dt}$

It is defined for a point of time .

Average velocity and Average speed

$\to$ Average velocity for a moving body in certain time interval may be zero, +ve or -ve but average speed can never be zero or –ve .

$\to$ Average speed ≥ Average velocity

$\to \dfrac{\text{Average speed}}{\text{Average velocity}} ≥1$

Average velocity

When a body moving along straight line covers distances $s_1 , s_2 , . . . s_n$ at different velocities $v_1 , v_2, ... v_n$ then

Average velocity = $\dfrac{\text{Total displacement}}{\text{Total time}}$

$V_{av}= \dfrac{(s_1 +s_2+ , . . .+ s_n)}{(t_1 + t_2+ .. . . .+ t_n)}$

$V_{av}= \dfrac{(s_1 +s_2+ , . . .+ s_n)}{(\dfrac{s_1}{v_1} + \dfrac{s_2}{v_2}+ .. . . .+ \dfrac{s_n}{v_n})}$

$\dfrac{(s_1 +s_2+ , . . .+ s_n)}{V_{av}} = \dfrac{s_1}{v_1} + \dfrac{s_2}{v_2}+ .. . . .+ \dfrac{s_n}{v_n}$

Equal Distance

If $s_1=s_2 . . . . . =s_n$