A bungee jumper makes a jump into the Gorge du Verdon in southern France from a platform 182 m above the bottom of the gorge. The jumper weighs 740 N and comes within 66 m of the bottom of the gorge. The cord's unstretched length is 30.0 m.
(a) Assuming that the bungee cord follows Hooke's law when it stretches, find its spring constant. [Hint: The cord does not begin to stretch until the jumper has fallen 30.0 m.](in N/m)
(b) At what speed is the jumper falling when he reaches a height of 95 m above the bottom of the gorge?(in m/s)
what is the formula?
To solve this problem, we can use the concepts of potential energy and Hooke's law.
(a) The potential energy of the bungee jumper can be found using the formula:
Potential energy = mgh
Where:
m = mass of the jumper (we'll assume it to be in kg for consistency)
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height above the bottom of the gorge
Since the weight (W) of the jumper is equal to the force due to gravity (W = mg), we can rewrite the potential energy equation as:
Potential energy = Wh
The maximum potential energy is when the jumper is at the highest point of the jump, which is 182 m above the bottom of the gorge. At this point, the length of the bungee cord is its unstretched length plus the distance fallen by the jumper (30 m + 182 m = 212 m). So, the potential energy can be written as:
Potential energy = W * (length of the bungee cord) = W * 212 m
Setting the potential energy equal to the gravitational potential energy (mgh), we can solve for the spring constant (k) using Hooke's law:
Potential energy = mgh = k * (extension of the bungee cord)^2
Simplifying, we get:
k = (mgh) / (extension of the bungee cord)^2 = (mgh) / (212 m - 30 m)^2
Now we can substitute the given values:
m = 740 N / 9.8 m/s^2 (to convert from N to kg)
g = 9.8 m/s^2
h = 182 m
extension of the bungee cord = 182 m - 30 m = 152 m
Plugging in these values, we can calculate the spring constant:
k = (740 / 9.8) * (9.8 * 182) / (152)^2 ≈ 250 N/m
Therefore, the spring constant of the bungee cord is approximately 250 N/m.
(b) To calculate the speed of the jumper when he reaches a height of 95 m above the bottom of the gorge, we can use the principles of conservation of mechanical energy.
When the jumper is at a height of 95 m above the bottom, the potential energy is given by:
Potential energy = mgh = m * 9.8 * 95
At this point, the kinetic energy (KE) of the jumper is equal to the potential energy, as there is no other form of energy present:
Kinetic energy = Potential energy = mgh
The kinetic energy of an object is given by the formula:
Kinetic energy = (1/2)mv^2
Setting the two expressions for kinetic energy equal, we can solve for the speed (v) of the jumper:
(1/2)mv^2 = mgh
Simplifying:
v^2 = 2gh
Substituting the given values:
g = 9.8 m/s^2
h = 95 m
v^2 = 2 * 9.8 * 95
Taking the square root of both sides:
v ≈ √(2 * 9.8 * 95) ≈ 43 m/s
Therefore, when the jumper reaches a height of 95 m above the bottom of the gorge, the speed is approximately 43 m/s.
To solve this problem, we can use the concept of potential energy and Hooke's law.
(a) To find the spring constant of the bungee cord, we can start by considering the potential energy at the maximum stretch of the cord (when the jumper is 66 m above the bottom of the gorge). At this point, the potential energy is equal to the weight of the jumper times the distance fallen:
Potential Energy = mgh
where:
m = mass of the jumper (given as 740 N/g, where g is the acceleration due to gravity, approximately 9.8 m/s^2)
h = height fallen (182 m - 66 m = 116 m)
Substituting the given values, we have:
Potential Energy = (740 N/g)(9.8 m/s^2)(116 m)
Next, we can relate the potential energy to the energy stored in the bungee cord using Hooke's law. According to Hooke's law, the potential energy is also equal to the energy stored in the cord, which can be calculated as:
Energy stored = (1/2)kx^2
where:
k = spring constant of the cord (what we want to find)
x = maximum stretch of the cord (116 m - 30 m = 86 m)
Setting the two expressions for potential energy equal, we have:
(740 N/g)(9.8 m/s^2)(116 m) = (1/2)k(86 m)^2
Now, we can solve for k:
k = [(740 N/g)(9.8 m/s^2)(116 m)] / [(1/2)(86 m)^2]
(b) To find the speed of the jumper when he reaches a height of 95 m above the bottom of the gorge, we can use the principle of conservation of mechanical energy. At that height, the potential energy of the jumper is equal to his kinetic energy:
Potential Energy = Kinetic Energy
So we have:
mgh = (1/2)mv^2
where:
m = mass of the jumper (given as 740 N/g)
g = acceleration due to gravity (approximately 9.8 m/s^2)
h = height above the bottom of the gorge (182 m - 95 m = 87 m)
v = velocity or speed of the jumper at that height
Simplifying the equation and solving for v, we get:
v = sqrt(2gh)
Plugging in the given values, we have:
v = sqrt(2 * 9.8 m/s^2 * 87 m)
25.2 n/m
24.2 m/s2