Question

Derive the following equations for uniformly accelerated motion: 

(i) v = u + at 

(ii) `"S = ut" + 1/2 "at"^2`

(iii) v² = u² + 2aS

where the symbols have their usual meanings.

Answer 1

Derivation of equations of motion

First equation of motion: 

Consider a particle moving along a straight line with uniform acceleration 'a'. At t = 0, let the particle be at A and u be its initial velocity, and at t = t, let v be its final velocity.

Acceleration = Change in velocity/Time a = (v - u)/t at = v - u

v = u+ at ... First equation of motion.

Second equation of motion: Average velocity = Total distance traveled/Total time taken Average velocity = s/t ...(1)

Average velocity can be written as (u+v)/2 Average velocity = (u+v)/2 ...(2)

From equations (1) and (2) s/t = (u+v)/2 ...(3)

The first equation of motion is v = u + at.

Substituting the value of v in equation (3), we get

s/t = (u + u + at)/2 s = (2u + at) t/2 = 2ut + at²/2 = 2ut/2 + at²/2

s = ut + (1/2) at² …Second equation of motion.

Third equation of motion: The first equation of motion is v = u + at. v - u = at ... (1)

Average velocity = s/t ...(2)

Average velocity =(u+v)/2 ...(3)

From equation (2) and equation (3) we get,

(u + v)/2 = s/t ...(4)

Multiplying eq (1) and eq (4) we get,

(v - u)(v + u) = at × (2s/t) (v - u)(v + u) = 2as

[We make the use of the identity a² - b² = (a + b) (a - b)]

v² - u² = 2as ...Third equation of motion.