A daredevil wishes to bungee-jump from a hot-air balloon 70.5 m above a carnival midway.?
He will use a piece of uniform elastic cord tied to a harness around his body to stop his fall at a point 15.0 m above the ground. Model his body as a particle and the cord as having negligible mass and a tension force described by Hooke's force law. In a preliminary test, hanging at rest from a 5.00-m length of the cord, the jumper finds that his body weight stretches it by 1.65 m.
He will drop from rest at the point where the top end of a longer section of the cord is attached to the stationary balloon.
(a) What length of cord should he use?
(b) What maximum acceleration will he experience?
- WhomeLv 78 months agoFavorite Answer
Use work energy theory.
Assume the balloon is stationary during the event.
from the preliminary test, we can find the spring constant k by summing forces to zero.
kx - mg = 0
kx = mg
k = mg/x
k = mg/1.65
as this was a 5 meter cord when relaxed, we can determine a k' for a unit length of cord
k' = 5(mg/1.65)
The maximum distance of fall desired is
70.5 - 15.0 - 55.5 m
during the jump, the potential energy will convert to spring potential
PE = PS
mgh = ½kx²
if the initial length of cord is L
x = 55.5 - L
and our spring constant will be
k = k'/L
55.5mg = ½(k' / L)(55.5 - L)²
55.5mg = ½(3.03mg) / L)(55.5 - L)²
55.5 = ½(3.03) / L)(55.5 - L)²
36.63 = (1 / L)(55.5 - L)²
36.63L = 3,080.25 - 111L + L²
0 = 3,080.25 - 147.63L + L²
L = (147.63 ±√(147.63² - 4(1)(3080.25))) / (2(1))
L = 25.148 m
L = 122.5 m which we ignore as it cannot function in this scenario.
L = 25.1 m ANSWER
With the length of cord set, the stretch will be
x = 55.5 - 25.1 = 30.4 m
The force applied by the cord at max stretch is
F = kx
F = (k'/25.1)(30.4)
F = (5(mg/1.65)/25.1)(30.4)
F = 3.67mg
so the maximum acceleration experienced by the jumper will be
a = F/m
a = 3.67 mg / m
a = 3.67 g's
while an observer on the ground will see the jumper's body appear to accelerate upward at one g less or
a = 2.67 g or about 26.2 m/s²
I hope this helps
Please select as Best Answer if it does.
- audreyLv 78 months ago
Ok. Does he want to win a teddy bear? What's your question?
- JetDocLv 78 months ago
Yes... AND? What's your question?