1
AMS 361: Applied Calculus IV by Prof Y. Deng
Homework 5
Assignment Date:
Wednesday (0
3
/11/2009)
Collection Date:
Wednesday (03/18/2009)
Grade:
Each problem is worth 10 points.
Problem 5.1 (Prob. 27, P. 107, Modified)
Suppose that a projectile is fired straight upward from
the surface of the earth with initial velocity
0
2
/
v
GM
R
. Then its height
y(t)
above the
surface satisfies the initial value problem
2
2
2
(
)
d y
GM
dt
y
R
,
(0)
0
y
,
0
(0)
y
v
. Compute the maximum altitude the projectile
reaches and the time (from launch) for the projectile to reach the maximum altitude.
Solution:
The maximum altitude
(
?
?𝑎?
)
:
?
2
?
??
2
=
−𝐺?
(
?
+
𝑅
)
2
;
?
=
??
??
;
?
2
?
??
2
=
??
??
⋅
??
??
=
?
(
??
??
)
?
??
??
=
−𝐺?
(
?
+
𝑅
)
2
?
?
0
??
=
−𝐺?
(
?
+
𝑅
)
2
??
?
0
?
(
?
)
2
2
−
?
0
2
2
=
𝐺?
? ?
+
𝑅
−
𝐺?
𝑅
=>
?
(
?
)
2
=
?
0
2
−
2
𝐺?? ?
𝑅
(
? ?
+
𝑅
)
? ?
= 0 =>
?
?𝑎?
=
𝑅
(
2
𝐺?
2
𝐺?−𝑅?
0
2
−
1)
The time
𝑇
??
(from launch) for the projectile to reach the maximum altitude:
??
??
=
?
=
?
0
2
−
2
𝐺?? ?
𝑅
(
? ?
+
𝑅
)
,
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2
1
?
?
?𝑎?
0
??
=
??
𝑇
??
0
, where T is the time the projectile going up.
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 Spring '08
 Staff
 Velocity, maximum altitude, Prof Y. Deng

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