Motion In A Straight Line

Motion In A Straight Line

Speed: The distance travelled by the body in unit time is called its speed.
        Speed (V) = \( \frac{{Dis\tan cetravelled}} {{Timetaken}} \)
*  speed is a scalar quantity.         
*  It is represented by v or u
units: CGS Unit : cm/s, SI unit: m/s
UNIFORM MOTION
If an object moving along the straight line covers equal displacement in equal intervals of time, i.e object moving with uniform velocity is said to be in uniform motion along a straight line. The following Fig. shows the position time graph of such a motion.

UNIFORM SPEED
If a particle moving along a straight line (say x-axis) travels equal distances in equal intervals of time however small the intervals may be, then the particle is moving with uniform speed.
NON- UNIFORM SPEED OR VARIABLE SPEED
If a particle moving along a straight line travels unequal distances in equal intervals of time or equal distances in unequal intervals of time, then it is said to be moving with nonuniform speed.
Eg: i) Motion of a freely falling body.
Eg: ii) Motion of a body thrown vertically upwards
AVERAGE SPEED
For a particle in motion (uniform or nonuniform), the ratio of total distance travelled to the total time of motion is called average speed.
\( \text{Average speed } = \frac{{\text{Total distance travelled}}} {{\text{Total time}}} \)        
\( = \frac{{x_2 - x_1 }} {{t_2 - t_1 }} \)
Where \( x_1 \) and \( x_2\) are the positions of an object at the instants  and  respectively.
    If \( S_1 ,S_2 ,S_3 ......S_n \) are the distances travelled by a particle in the time intervals \( t_1 ,t_2 ,.....t_n \)...... respectively then,
            Average Speed=\( \frac{{S_1 + S_2 + S_3 + ...... + S_n }} {{t_1 + t_2 + ,.... + t_n }} \)
 INSTANTANEOUS SPEED
The speed of a particle at a particular instant of time is called it's instantaneous speed. (or)
It is also defined as the limit of average speed as the time interval (\( \Delta t \)) becomes infinitesimally small.
If \( \Delta x \) is the distance travelled by a particle in a time interval \( \Delta t \) then
            Speed=\( \frac{{\Delta x}} {{\Delta t}} \)
If the time interval  is chosen to be very small, i.e., as \( \Delta t \to 0 \), then the corresponding speed is called instantaneous speed.
          \( \therefore \mathop {\lim }\limits_{\Delta t \to 0} \frac{{\Delta x}} {{\Delta t}} = \frac{{dx}} {{dt}} \)  =instantaneous speed
 Non - uniform speed: If a body travels unequal distances in equal intervals of time (or)equal distance in unequal intervals of time the body is said to be travelling with non uniform (or) variable speed.
Application-1:
A body travelling between two positions travels first half of the distance with speed \( v_1 \), and the next half of the distance with speed \( v_2\)
The  average speed of total motion is \( \frac{{2v_1 v_2 }} {{v_1 + v_2 }} \)
Let x be the total distance between two positions.
Let  be the time for first half and be the time for the next half of the distance
\( \text{Average speed } = \frac{{\text{Total distance travelled}}} {{\text{Total time}}} \)
\( \frac{S} {{t_1 + t_2 }} = \frac{S} {{\frac{S} {{2v_1 }} + \frac{S} {{2v_2 }}}} = \frac{{2v_1 v_2 }} {{v_1 + v_2 }} \)

Application-2:
A body is travelling between two positions A, B. It travels from A to B with speed \( v_1 \) and then from B to A in the same path with speed \(v_2\). The average speed of total motion is \( \frac{{2v_1 v_2 }} {{v_1 + v_2 }} \)

Application-3:
A body is travelling between two positions. The total distance is divided into n equal parts. These parts are travelled with speeds \( v_1 ,v_2 ,.....v_n \) respectively. The average speed of total motion is such that
        \( \frac{n} {{\text{Average speed }}} = \frac{1} {{v_1 }} + \frac{1} {{v_2 }} + .... + \frac{1} {{v_n }} \)

Application-4:
A body travelling between two positions travels with speed \( v_1 \) for time \( t_1 \) and then with speed \( v_2 \) for time \( t_2 \) For the total motion,
        \( \text{Average speed } = \frac{{v_1 t_1 + v_2 t_2 }} {{t_1 + t_2 }} \)

Application-5:
A body travelling between two positions travels first half of the time with speed \( v_1 \) and the next half of the time with speed \( v_2 \). The average speed of total motion is
        =\( \frac{{v_1 + v_2 }} {2}\)
Application-6:
A body travelling between two positions travels  for the time intervals \( t_1 ,t_2 ,.....t_n \) with speeds \( v_1 ,v_2 ,.....v_n \) respectively.
\( \text{Average speed } = \frac{{\text{Total distance travelled}}} {{\text{Total time}}} \)
=\( \frac{{v_1 t_1 + v_2 t_2 + ..... + v_n t_n }} {{t_1 + t_2 + ,.... + t_n }} \)