5.3 Solutions
203
v 2
2
=
G M
x
+ C
Initially v = 0, x = r
∴ C = −
G M
r
∴ v =
dx
dt
=
√
2G M
1
x
−
1
r
dt =
1
√
2G M
dx
1
x
−
1
r
Integrating
t =
dt =
1
√
2G M
r
0
dx
1
x
−
1
r
Put x = r cos
2
θ, dx = −2r sin θ cos θ dθ
t = −2
r 3
2G M
0
π/2
cos
2
θ d θ = 2
r 3
2G M
θ
2
+
sin 2θ
4
π/2
0
=
π
2
√
2
r 3
G M
But the period of earth’s orbit is
T = 2π
r 3
G M
∴ t =
T
4
√
2
=
365
4
√
2
= 64.53 days
Fig. 5.9
5.5 If the earth were at rest, then the gravitational force on a body of mass at P
would be in the direction PO, i.e. towards the centre of the earth, Fig. 5.9.
203
v 2
2
=
G M
x
+ C
Initially v = 0, x = r
∴ C = −
G M
r
∴ v =
dx
dt
=
√
2G M
1
x
−
1
r
dt =
1
√
2G M
dx
1
x
−
1
r
Integrating
t =
dt =
1
√
2G M
r
0
dx
1
x
−
1
r
Put x = r cos
2
θ, dx = −2r sin θ cos θ dθ
t = −2
r 3
2G M
0
π/2
cos
2
θ d θ = 2
r 3
2G M
θ
2
+
sin 2θ
4
π/2
0
=
π
2
√
2
r 3
G M
But the period of earth’s orbit is
T = 2π
r 3
G M
∴ t =
T
4
√
2
=
365
4
√
2
= 64.53 days
Fig. 5.9
5.5 If the earth were at rest, then the gravitational force on a body of mass at P
would be in the direction PO, i.e. towards the centre of the earth, Fig. 5.9.