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Question
Derive the equations
(i) v= u+at and
(ii) v2-u2= 2as, where the symbols have their usual meanings.
Solution
This is graph plotted between velocity and time.
(i) The initial velocity of the body is u at t=O. The velocity of the body increases at a uniform rate and this increase in velocity up to time t is depicted by a straight line PQ. The slope of line PQ gives acceleration a.
a= QR/ PR.
PR= OS =t
SR= OP =u
QR= a x PR.
= a x t .
The point Q corresponds to the final velocity v after time t .
v = QR+ RS and generally we write v = u + at .
This is first equation of motion.
(ii) The area enclosed under a velocity time curve gives the distance covered by a moving body. So total distance S covered by a uniformly accelerating body is given by area of trapezium OSQP.
S = area of trapezium OSQP.
AREA of rectangle OSRP + area of triangle PRQ.
S = OP x OS + 1/2 PR xQR.
s = u x t + 1/2 x t x at .
S= ut + 1/2 at2
This is known as second equation of motion.
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