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CBSE Class 9 Answered

how is equation if motion is formed in graphical method

Asked by poornimaram165 | 21 Jun, 2022, 09:16: PM
Expert Answer
Let us consider motion is one dimensional with uniform acceleration.
 
In uniform accelrated motion, velocity increases linearly with time.
 
Velocity v at a time t will be
 
v = u + k t
 
where u is intial velocity at t = 0 and k is constant
 
Velocity that varies linearly with respect to time is plotted in above figure.
 
Velocity is u at t = 0  and velocity is v at time t
 
slope = dv/dt = (v-u) / t
 
dv/dt = rate of change of velocity with respect to time = acceleration a
 
Hence we get ,  acceleration a = ( v - u ) / t
 
from above expression we get , v = u + ( a t ) ....................... (1)
 
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In an uniform accelerated motion, let u be initial velocity at time t1 and v be velocity at later time t2 .
 
Graph plotted above shows the variation of velocity with respect to time along the line AB .
 
In velocity-time graph , distance travelled from time t1 to time tis area under the line AB bounded by time axis as shown in figure.
 
Area ABCD under the line AB is a trapezium .
 
Area of trapezium ABCD = (1/2) ( AC + BD ) × CD = (1/2) ( u + v ) (t1 - t2 )
 
Hence distance S travelled between time t1 and time t2 is given as
 
S = (1/2) ( u + v ) (t1 - t2 )  ..................... (2)
 
If we substitute v from eqn.(1) and consider time duration t = ( t1 - t2 ) , then we have
 
S = (1/2) [ u + u + (a t ) ] × t
 
Hence , we get ,  S =  ( u t ) + [ (1/2) a t2 ] .........................(3)
 
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Let us substitute time duration t = ( t1 - t2 ) in eqn.(2) using eqn.(1) as
 
t = ( v - u ) / a
 
Then we have
 
S = (1/2) ( u + v ) [ (v-u) / a ]
 
By simplifying above expression, we get
 
v2 = u2 + ( 2 a S )  ..................................................(4)
 
Eqn.(1) , (3) and (4) are known as equation of motion in one dimension with uniform acceleration
 
Answered by Thiyagarajan K | 21 Jun, 2022, 10:37: PM
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