# (a) Derive the formula: s = ut + ½ at² where the symbols have usual meanings. (b) A train starting from stationary position and moving with uniform acceleration attains a speed of 36 km per hour in 10 minutes. Find its acceleration.

By BYJU'S Exam Prep

Updated on: September 13th, 2023

(a) We have to prove that

s = ut + 1/2 at2

Where

s is the distance

u is the initial velocity

a is the acceleration

t is the time

Proof –

Acceleration = (Final velocity – Initial velocity)/t

a = (v – u)/ t

at = v – u …. (1)

We know that

Distance traveled = average velocity x time

s = (v + u)/ 2 x t

Using equation (1)

s = (u + at + u)/2 x t

s = [(2u + at)/ 2] x t

s = ut + 1/2 at2

(b) If a train starts from a stationary position, initial velocity = 0

Uniform acceleration refers to constant acceleration

It is given that

Final velocity V = 36 km/h = 36 x 5/18 = 10 m/s [As 1km/hr = 5/18 m/s]

Time taken t = 10 minutes= 10 x 60 = 600 seconds

From the kinematics equation

V = u + at

10 = 0 + a x 600

a = 10/600

a = 1/60 m/s2

a = 0.1667 x 10⁻1 m/s2 = 1.67 x 10⁻2 m/s2

Therefore, the formula is derived and the acceleration is 1.67 x 10⁻2 m/s2.

Summary:

## (a) Derive the formula: s = ut + 1/2 at2 where the symbols have usual meanings. (b) A train starting from a stationary position and moving with uniform acceleration attains a speed of 36 km per hour in 10 minutes. Find its acceleration.

(a) The formula: s = ut + 1/2 at2 where the symbols have usual meanings is derived

(b) A train starting from a stationary position and moving with uniform acceleration attains a speed of 36 km per hour in 10 minutes. Its acceleration is 1.67 x 10⁻2 m/s2.

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