A 72 kg bungee jumper jumps from a bridge. She is tied to a bungee cord whose unstretched length is 12 m , and falls a total of 35 m . Calculate the maximum acceleration she experiences.
It has to be at max stretch...
f=kx
mg=k(35-12)=23k
so you have k=mg/23
max acceleration= maxforce/m-g
= g/(23) -g
= g(1/23)-g
=-22/23 g negative means upward
To calculate the maximum acceleration experienced by the bungee jumper, we can utilize the following equation:
acceleration = (change in velocity) / (time taken)
First, let's calculate the change in velocity. To do this, we need to determine the final velocity (vf) and the initial velocity (vi).
The final velocity can be found using the equation:
vf^2 = vi^2 + 2 * acceleration * distance
In this case, the distance is the total fall of 35 m, the initial velocity is 0 m/s (since the jumper starts from rest), and the final velocity is the velocity just before the bungee cord starts to stretch.
Thus, the equation becomes:
vf^2 = 0 + 2 * acceleration * 35 m
Next, we know that when the bungee cord starts to stretch, the jumper starts decelerating until they come to a stop momentarily. At this point, the bungee cord reaches its maximum stretching length and begins to accelerate the jumper in the opposite direction.
Using the equation for distance traveled during constant acceleration:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
The distance traveled while decelerating is given by:
12 m = (0 * time) + (0.5 * acceleration * time^2)
Simplifying, we have:
6 m = 0.5 * acceleration * time^2
Similarly, the distance traveled while accelerating is given by:
23 m = (vf * time) + (0.5 * acceleration * time^2)
Now, we can substitute the value of vf^2 from the previous equation into the equation above:
23 m = (sqrt(2 * acceleration * 35) * time) + (0.5 * acceleration * time^2)
Simplifying and combining the two equations, we get:
29 m = 0.5 * acceleration * time^2
Next, we can eliminate time from the two equations by isolating it in either equation and substituting it into the other.
Let's solve the first equation for time:
6 m = 0.5 * acceleration * time^2
12 m = acceleration * time^2
time^2 = 12 m / acceleration
time = sqrt(12 m / acceleration)
Now, let's substitute this value of time into the second equation:
23 m = (sqrt(2 * acceleration * 35) * sqrt(12 m / acceleration)) + (0.5 * acceleration * (sqrt(12 m / acceleration))^2)
Simplifying:
23 m = sqrt(70 m * acceleration) * sqrt(12 m / acceleration) + 6 m
Canceling out the square roots:
23 m = sqrt(840 m^2 / acceleration)
Squaring both sides:
(23 m)^2 = 840 m^2 / acceleration
529 m^2 = 840 m^2 / acceleration
Simplifying:
acceleration = 840 m^2 / (529 m^2)
acceleration = 1.59 m/s^2
Therefore, the maximum acceleration experienced by the bungee jumper is approximately 1.59 m/s^2.