26
1 Kinematics and Statics
V 2 + u 2
V 2
= e
2 gh
V 2
h =
V 2
2g
ln
1 +
u 2
V 2
(6)
1.25 The particle reaches the height h given by
h =
V 2
2g
ln
1 +
u 2
V 2
(by prob. 1.24)
The velocity at any point during the descent is given by
v
2
= V
2
1 − e
−
2 gx
V 2
(by prob. 1.23)
The velocity of the body when it reaches the point of projection is found by
substituting h for x:
∴ v
2
= V
2
1 −
V 2
V 2 + u 2
=
u 2 V 2
V 2 + u 2
Loss of kinetic energy =
1
2
mu
2
−
1
2
mv
2
=
1
2
mu
2
1 −
V 2
V 2 + u 2
=
1
2
mu
2
u 2
V 2 + u 2
1.3.3 Motion in Two Dimensions
1.26 (i)
dx
dt
= 6 + 2t
dx = 6
dt + 2
tdt
x = 6t + t
2
+ C
x = 0, t = 0; C = 0
x = 6t + t
2
dy
dt
= 4 + t
dy = 4
dt +
tdt