The Royal Gorge bridge over the Arkansas River is 310 m above the river. A 64 -k

Question

The Royal Gorge bridge over the Arkansas River is 310 m above the river. A 64-kg bungee jumper has an elastic cord with an unstressed length of 59 m attached to her feet. Assume that, like an ideal spring, the cord is massless and provides a linear restoring force when stretched. The jumper leaps, and at at her lowest point she barely touches the water. After numerous ascents and descents, she comes to rest at a height h above the water. Model the jumper as a point particle and assume that any effects of air resistance are negligible.

h=150 m.

Find the maximum speed of the jumper.

Explanation / Answer

Using conservation of

mechanical energy, express her

total energy E:

E = K +Ug +Us = Ei = 0

Because v is a maximum when K is

a maximum, solve for K to obtain:

( ) 2

2

1

g s

mg d x kx

K U U

= + ?

= ? ? (2)

Use the condition for an extreme

value to obtain:

= mg ?kx = 0 dx

dK

?

k

mg x =

Substitute numerical values and

evaluate x:

( )( ) 109m

5.40N/m

60kg 9.81m/s2

x = =

From equation (2) we have: ( ) 2

2

2 1

2

1 mv = mg d + x ? kx

Solve for v to obtain: ( )

m

kx

v g d x

2

= 2 + ?

Substitute numerical values and evaluate v for x = 109 m:

( )( ) ( )( ) 45m/s

60kg

5.4N/m 109m 2 9.81m/s 50m 109m

2

2 v = + ? =

Because 0, 2

2

= ?k <

dx

d K

x = 109 m corresponds to Kmax and so v is a maximum

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