(i) Velocity-Time Equation(v = u + at)
Consider an object moving along a straight line with uniform acceleration
a'. Let u be the initial velocity i.e., velocity at t = 0. Let v be the
final velocity i.e., velocity at time t. If dv is the infinitesimally
small change in velocity in infinitesimally small time-interval dt, then
acceleration, a = dv/dt
or dv = a dt
Let us integrate it within the conditions of motion (that is, when
't' = 0, 'v' = u and when 't' = t, 'v' = v).
∫ dv =
∫ a dt
| v |
= a |
t |
v - u = a(t-0)
v = u + (at)
Derivation of v = v 0 + (at)
Here v 0 is the velocity at 't' = 0.
∫ v
v 0
dv =
∫ a dt
v - v 0 = at or v = v 0 + (at)
(ii) Displacement-Time Equation (S = (ut) + [(1/2)(at 2
)] )
Consider an object moving along a straight line with uniform
acceleration 'a'. Let u be the initial velocity of the object. Let S
be the displacement of the object in time t. Let, at an instant, dS be
the infinitesimally small displacement of the object in
infinitesimally small time-interval dt. The instantaneous velocity
v is given by
v = dS/dt or dS = v dt
But v = u + (at)
∴ dS = [u + (at)] dt
Let us integrate it within the conditions of motion (that is, when
t=0, S=0 and when 't' = t, 'S' = S).
∫ dS =
∫ [u + (at)]dt
|
S | = u
∫ dt + a
∫ t dt
S - 0 = u |
t |
+ a |
t 2 /2 |
S = [u(t-0)] + [(1/2)a(t 2 -
0)]
S = (ut) + [(1/2)at 2 ]
Derivation of Position-Time Equation (x = x 0 +
(v 0 t) + [(1/2)at 2 ])
With reference to the origin of position-axis, let x 0 be the
displacement at 't' = 0. Let x be the displacement at time t. Let
v 0 be the velocity at 't' = 0.
The instantaneous velocity of the object is given by
v = dx/dt or dx = v dt
or dx = v 0 dt + (at) dt [ ∵ v = v 0 +
(at)]
Integrating within conditions of motion,
∫ x
x 0
dx =
∫ v 0 dt
+ ∫ (at) dt
| x
| x
x 0
= v 0
| t
| + a
| t 2 /2
|
x - x 0 = v 0 (t-0) + [(1/2)a(
t 2 - 0)]
or x = x 0 + (v 0 t) +
[(1/2)at 2 ]
(iii) Velocity-Displacement Equation (v 2 -
u 2 = 2aS)
Consider an object moving along a straight line with uniform acceleration
'a'.
Then a = dv/dt
or a = (dv/dS) . (dS/dt) (Chain rule)
or a = v dv/dS
or v dv = a dS
Let u be the velocity of the object at start, that is, at S = 0. Let v
be the velocity at a distance S from the starting point.
Integrating within the conditions of motion, we get
∫ v dv =
∫ a dS
| v 2
/2 | = a
| S
|
or (v 2 /2) - (u 2 /2)
= a(S-0)
or v 2 - u 2 = 2aS
Derivation of v 2 - v 0 2
= 2a(x - x 0 )
a = dv/dt = (dv/dx) . (dx/dt)
a = v(dv/dx) or v dv = a dx
Integrating within the conditions of motion,
∫ v
v 0
v dv =
∫ x
x 0
a dx
|
v 2 /2 |
v
v 0
= a |
x |
x
x 0
v 2 - v 0 2
= 2a (x - x 0 )
(iv) Equation for displacement in nth second of motion
[S nth = u + [(a/2)(2n-1)] ]
We know that v = dS/dt or dS = v dt
or dS = [u + (at)] dt [ ∵ v = u + (at)]
or dS = u dt + (at) dt
When t = n-1, then S = S n-1
When t = n, then S = S n
Integrating within the conditions of motion,
∫ S n
S n-1
dS =
∫ n
n-1
u dt +
∫ n
n-1
(at) dt
| S
| S n
S n-1
= u
| t
| n
n-1 + a
|
t 2 /2 |
n
n-1
or S n - S n-1
= u[n-(n-1)] + (a/2)[n 2 - (n-1) 2 ]
or S nth = u + [(a/2)(2n-1)]
S nth represents displacement of the object in
nth second of motion.
Derivation of x nth = v 0 + [(a/2)(
2n-1)]
v = dx/dt or dx = v dt
or dx = [v 0 + (at)] dt
or dx = v 0 dt + (at) dt
Integrating within the conditions of motion,
∫ x n
x n-1
dx =
∫ n
n-1
v 0
dt + ∫ n
n-1
(at) dt
| x
| x n
x n-1 = v 0
| t
| n
n-1 + a
|
t 2 /2 |
n
n-1
or x n - x n-1
= v 0 [n - (n-1)] + {(a/2)[n 2 -
(n-1) 2 ]}
or x nth = v 0 + [(a/2)(2n-1)]
The kinematic equations of motion in vector
form are:
(i)
v
→
=
u
→ +
a →
t
(ii)
S →
= u
→
t + [(1/2)( a
→
t 2 )]
(iii)
S
→
nth =
u
→ +
[( a
→
/2)(2n-1)]
(iv)
( v
→
. v
→
) - ( u
→
. u
→
) = 2 a
→
. S
→
While the first three equations
are vector equations, the fourth equation is a scalar equation.
If acceleration and velocity are not collinear, then vector
equations are useful.