Question

1. Derive the equations v = u + at, s = ut + ½ at2 and v2 = u2 + 2as graphically.

Answer 1

To derive : The equations ....

1. v = u + at
2. s = u + 1/2 at²
3. v² = u² + 2as

solution : let a particle moves with initial velocity u after time t, its velocity becomes v due to acceleration acting on particle is a.

see figure,

slope of velocity - time graph = acceleration

⇒(v - u)/(t - 0) = a

⇒v - u = at

⇒v = u + at .........(1)

area enclosed the velocity - time graph = displacement covered by particle

⇒area of trapezium formed as shown in figure = S

⇒S = 1/2 [v + u ] × t

from equation (1),

⇒S = 1/2 [u + at + u ] × t

⇒S = 1/2 [2u + at] × t

⇒S = ut + 1/2 at² ...........(2)

we know, acceleration, a = v dv/ds

⇒a ∫ds = ∫v dv

⇒a[s] = [v²/2]

⇒as = 1/2 [v² - u²]

⇒2as = v² - u²

⇒v² = u² + 2aS ............(3)

Answer 2

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