derive equations of motion for an object under uniformly accelerated motion
Answers
There are three equations of bodies moving with uniform acceleration which we can use to solve problems of motion
First Equation of motion
The first equation of motion is
v
=
u
+
a
t
v=u+at , where v is the final velocity and u is the initial velocity of the body.
First equation of motion gives velocity acquired by body at any time
t
t.
Now we know that acceleration
equation of acceleration
so,
a
=
v
−
u
t
a=v−ut
and,
a
t
=
v
−
u
at=v−u
rearranging above equation we get first equation of motion that is
v
=
u
+
a
t
v=u+at
Second Equation of motion
Second equation of motion is
s
=
u
t
+
1
2
a
t
2
s=ut+12at2
where
u
u is initial velocity,
a
a is uniform acceleration and
s
s is the distance travelled by body in time
t
t.
Second equation of motion gives distance travelled by a moving body in time
t
t.
To obtain second equation of motion consider a body with initial velocity
u
u moving with acceleration a for time
t
t its final velocity at this time be
v
v. If body covered distance
s
s in this time
t
t , then average velocity of the body would be
average velocity
Distance travelled by the body is
From first equation of motion
v
=
u
+
a
t
v=u+at
So putting first equation of motion in above equation we get ,
s
=
u
+
u
+
a
t
2
×
t
=
(
2
u
+
a
t
)
t
2
=
2
u
t
+
a
t
2
2
s=u+u+at2×t=(2u+at)t2=2ut+at22
Rearranging it we get
s
=
u
t
+
1
2
a
t
2
s=ut+12at2
Third equation of motion
Third equation of motion is
v
2
=
u
2
+
2
a
s
v2=u2+2as where
u
u is initial velocity,
v
v is the final velocity,
a
a is uniform acceleration and
s
s is the distance travelled by the body.
This equation gives the velocity acquired by the body in travelling a distance
s
s.
Third equation of motion can be obtained by eliminating time t between first and second equations of motion.
So, first and second equations of motion respectively are
v
=
u
+
a
t
v=u+at and
s
=
u
t
+
1
2
a
t
2
s=ut+12at2
Rearranging first equation of motion to find time t we get
t
=
v
−
u
a
t=v−ua
Putting this value of t in second equation of motion we get
s
=
u
(
v
−
u
)
a
+
1
2
a
(
v
−
u
a
)
2
s=u(v−u)a+12a(v−ua)2
s
=
u
v
−
u
2
a
+
a
(
v
2
+
u
2
−
2
u
v
)
2
a
2
s=uv−u2a+a(v2+u2−2uv)2a2
s
=
2
u
v
−
2
u
2
+
v
2
+
u
2
−
2
u
v
2
a
s=2uv−2u2+v2+u2−2uv2a
Rearranging it we get
v
2
=
u
2
+
2
a
s
v2=u2+2as
These three equations of motion are used to solve uniformly accelerated motion problems and following three important points should be remembered while solving problems
if a body starts moving from rest its initial velocity
u
=
0
u=0
if a body comes to rest i.e., it stops then its final velocity would be
v
=
0
v=0
If a body moves with uniform velocity then its acceleration would be zero.