Page 62 - Class-11-Physics-Part-1_Neat

48 PHYSICS

This equation can also be obtained by t
at
d
substituting the value of t from Eq. (3.6) into = ∫ (v 0 + ) t
0
Eq. (3.8). Thus, we have obtained three
important equations : 1 2
x – x = v t + a t
0
0
v = v + at 2
0
1 2
1 2 x = x + v t + a t
0
0
x = v t + at 2
0
2 We can write
v = v + 2ax (3.11a) a = dv = dv dx = v dv
2
2
0
dt dx dt dx
connecting five quantities v , v, a, t and x. These
0 or, v dv = a dx
are kinematic equations of rectilinear motion
for constant acceleration. Integrating both sides,
The set of Eq. (3.11a) were obtained by ∫ v v v = ∫ x
d
assuming that at t = 0, the position of the v 0 d x 0 a x
particle, x is 0. We can obtain a more general 2 2
equation if we take the position coordinate at t v – v 0 = a (x – x )
= 0 as non-zero, say x . Then Eqs. (3.11a) are 2 0
0
modified (replacing x by x – x ) to : 2 2
0 v = v + 2a (x – x 0 )
0
v = v + at The advantage of this method is that it can be
0
1 used for motion with non-uniform acceleration
x = x + v t + at 2
0 0 (3.11b) also.
2
Now, we shall use these equations to some
v = v + 2 (x − x 0 ) (3.11c) important cases. t
2
2
a
0
Example 3.4 A ball is thrown vertically
t
Example 3.3 Obtain equations of motion upwards with a velocity of 20 m s from
–1
t
for constant acceleration using method of the top of a multistorey building. The
calculus. height of the point from where the ball is
thrown is 25.0 m from the ground. (a) How
Answer By definition high will the ball rise ? and (b) how long
will it be before the ball hits the ground?
dv –2
a = Take g = 10 m s .
dt
dv = a dt
Integrating both sides Answer (a) Let us take the y-axis in the
vertically upward direction with zero at the
t
v
∫ dv = ∫ a dt ground, as shown in Fig. 3.13.
v 0 0 –1
Now v = + 20 m s ,
o

–2
=a ∫ t t d (a is constant) a = – g = –10 m s ,
0
v = 0 m s –1
v – v = at If the ball rises to height y from the point of
0
v = v + at launch, then using the equation
0 2 2
v = v + 2 a ( y – y 0 )
0
dx we get
Further, v =
dt 0 = (20) + 2(–10)(y – y )
2
dx = v dt 0
Integrating both sides Solving, we get, (y – y ) = 20 m.
0
∫ x dx = ∫ t v d t (b) We can solve this part of the problem in two
x 0 0 ways. Note carefully the methods used.
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