Question

derieve second and third equation of motion

Answer 1

Equation for distance –time relation

Distance covered by the object in the given time ‘t’ is given by the area of the trapezium ABDOE

Let in the given time, t the distance covered by the moving object = s

The area of trapezium, ABDOE

= Distance (s) = Area of △ABD+△ABD+ Area of ADOE

⇒s=12×AB×AD+(OD×OE)⇒s=12×AB×AD+(OD×OE)

⇒s=12×DC×AD+(u+t)⇒s=12×DC×AD+(u+t)

[Since, AB=DCAB=DC]

⇒s=12×at×t+ut⇒s=12×at×t+ut

⇒s=12×at×t+ut⇒s=12×at×t+ut

[∵ DC=atDC=at]

⇒s=12at2+ut⇒s=12at2+ut

⇒s=ut+12at2⇒s=ut+12at2

The above expression gives the distance covered by the object moving with uniform acceleration. This expression is known as second equation of motion.

Equation for Distance Velocity Relation: Third equation of Motion:

The distance covered by the object moving with uniform acceleration is given by the area of trapezium ABDO

Therefore,

Area of trapezium ABDOE

=12×(sum of parallel sides+distance between parallel sides)=12×(sum of parallel sides+distance between parallel sides)

⇒ Distance (s) =12(DO+BE)×OE=12(DO+BE)×OE

⇒s=12(u+v)×t⇒s=12(u+v)×t ----(iii)

Now from equation (ii) a=v−uta=v-ut

∴t=v−ut∴t=v-ut ----(iv)

After substituting the value of tt from equation (iv) in equation (iii)

⇒s=12(u+v)×(v−u)a⇒s=12(u+v)×(v-u)a

⇒s=12a(v+u)(v−u)⇒s=12a(v+u)(v-u)

⇒2as=(v+u)(v−u)⇒2as=(v+u)(v-u)

⇒2as=v2−u2⇒2as=v2-u2

⇒2as+u2=v2⇒2as+u2=v2

⇒v2=u2+2as⇒v2=u2+2as

The above expression gives the relation between position and velocity and is called the third equation of motion.

Answer 2

(2) Second equation of motion:
s = ut + 1/2 at^2

sol.
Let the distance travelled by the body be “s”.
We know that
Distance = Average velocity X Time

Also, Average velocity = (u+v)/2

.: Distance (t) = (u+v)/2 X t …….eq.(1)

Again we know that:
v = u + at
substituting this value of “v” in eq.(2), we get

s = (u+u+at)/2 x t
=>s = (2u+at)/2 X t
=>s = (2ut+at^2)/2
=>s = 2ut/2 + at^2/2

or s = ut +1/2 at^2

This is the 2nd equation of motion.
……………………………………………………………
(3) Third equation of Motion
v^2 = u^2 +2as

sol.
We know that
V = u + at
=> v-u = at
or t = (v-u)/a ………..eq.(3)

Also we know that
Distance = average velocity X Time
.: s = [(v+u)/2] X [(v-u)/a]
=> s = (v^2 – u^2)/2a

=>2as = v^2 – u^2

or v^2 = u^2 + 2as

This is the third equation of motion.