28
1 Kinematics and Statics
or
r 12 =
(u 1 + u 2 )
g
√
u 1 u 2
(4)
where we have used (2).
1.28 Consider the equation
s = ut +
1
2
at
2
(1)
Taking upward direction as positive, a = −g and let s = h, the height of the
tower, (1) becomes
h = ut −
1
2
gt
2
or
1
2
gt
2
− ut + h = 0
( 2 )
Let the two roots be t 1 and t 2 . Compare (2) with the quadratic equation
ax
2
+ bx + c = 0
( 3 )
The product of the two roots is equal to c/a. It follows that
t 1 t 2 =
2h
g
or
√
t 1 t 2 =
2h
g
= t 3
which is the time taken for a free fall of an object from the height h.
1.29 Let the shell hit the plane at p(x, y), the range being AP = R, Fig. 1.19. The
equation for the projectile’s motion is
y = x tan θ −
gx 2
2u 2 cos 2 θ
(1)
Now y = R sin α
(2)
x = R cos α
(3)
Fig. 1.19
1 Kinematics and Statics
or
r 12 =
(u 1 + u 2 )
g
√
u 1 u 2
(4)
where we have used (2).
1.28 Consider the equation
s = ut +
1
2
at
2
(1)
Taking upward direction as positive, a = −g and let s = h, the height of the
tower, (1) becomes
h = ut −
1
2
gt
2
or
1
2
gt
2
− ut + h = 0
( 2 )
Let the two roots be t 1 and t 2 . Compare (2) with the quadratic equation
ax
2
+ bx + c = 0
( 3 )
The product of the two roots is equal to c/a. It follows that
t 1 t 2 =
2h
g
or
√
t 1 t 2 =
2h
g
= t 3
which is the time taken for a free fall of an object from the height h.
1.29 Let the shell hit the plane at p(x, y), the range being AP = R, Fig. 1.19. The
equation for the projectile’s motion is
y = x tan θ −
gx 2
2u 2 cos 2 θ
(1)
Now y = R sin α
(2)
x = R cos α
(3)
Fig. 1.19