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 t2 is 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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