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001 (part 1 of 7) 2.0 points
AstaroFmass7
×
10
30
kg is located at
~
r
S
=
h
6
×
10
12
,
2
×
10
12
,
0
i
m
.
Ap
lane
to
Fma
s
s3
×
10
24
kg is initially lo
cated at
~
r
P
i
=
h
2
×
10
12
,
4
×
10
12
,
0
i
m
and is moving with an initial velocity oF
~
v
P
i
=
h
2000
,
16000
,
0
i
m
/
s
.
At a time 1
×
10
6
slater
,whatisthenew
velocity oF the planet? Start by fnding the
x
component,
v
P
f,x
.
Correct answer: 2020
.
88 m
/
s.
Explanation:
In order to use the Momentum Principle to
fnd the new velocity, we have to frst fnd the
net gravitational Force on the planet From the
star.
The Formula For the gravitational Force be
tween two objects is
~
F
gr
=

Gm
1
m
2

~
r

2
ˆ
r.
We know the masses oF the objects, so we just
need to fnd the magnitude oF the distance
between them and the unit vector pointing
From the star to the planet.
To fnd the vector pointing From the star to
the planet (let’s call it
~
r
P

S
), we subtract the
position oF the star From that oF the planet:
~
r
P

S
=
~
r
P

~
r
S
=
h
2
×
10
12
,
4
×
10
12
,
0
i
m
h
6
×
10
12
,
2
×
10
12
,
0
i
m
=
h
4
×
10
12
,
2
×
10
12
,
0
i
m
.
We need the length oF this vector and the
unit vector pointing in the same direction.
The length (or magnitude) is given by
±
±
±
~
r
P

S
±
±
±
=
q
(

4
×
10
12
m)
2
+(2
×
10
12
m)
2
=4
.
47214
×
10
12
m
.
And we use the magnitude to help us fnd
the unit vector:
ˆ
r
P

S
=
~
r
P

S

~
r
P

S

=
h
4
×
10
12
,
2
×
10
12
,
0
i
m
4
.
47214
×
10
12
m
=
h
0
.
894427
,
0
.
447214
,
0
i
.
Now we have everything we need to calcu
late the components oF
~
F
gr
.L
e
t
’
sg
oa
h
e
a
d
and fnd all three components using vector
algebra.
~
F
gr
=

Gm
S
m
P

~
r
P

S

ˆ
r
P

S
=

(
G
)(7
×
10
30
kg)(3
×
10
24
kg)
(4
.
47214
×
10
12
m)
2
×h
0
.
894427
,
0
.
447214
,
0
i
=
h
6
.
26412
×
10
19
,

3
.
13206
×
10
19
,
0
i
N
,
where
G
=6
.
67
×
10

11
m
3
·
kg

1
·
s

2
.
Now that we know the net gravitational
Force on the planet due to the star, we can use
the Momentum Principle,
Δ
~
p
=
~
F
net
Δ
t
=
~
F
gr
Δ
t,
to fnd the new velocity, using the velocity
update Formula we can derive From the Mo
mentum Principle:
~
p
P
f
=
~
p
P
i
+
~
F
gr
Δ
t
m
P
~
v
P
f
=
m
P
~
v
P
i
+
~
F
gr
Δ
t
~
v
P