A bungee jumper (m = 61.00 kg) tied to a 45.00 m cord, leaps off a 75.00 m tall bridge. He falls to 5.00 m above the water before the bungee cord pulls him back up. What is the magnitude of the impulse exerted on the bungee jumper as the cord stretches?
Can anyone show me how to do this step by step?
First, you need to find the velocity when the cord begins to stretch, i.e after the first 45m of free fall
Vf^2 = Vi^2 + 2ad
Vf^2 = (0) +(2)(-9.81)(-45)
Vf = 29.7m/s
Using this velocity you can find the momentum, and since Impulse in just the chanage in momentum, it will be the momentum at this point.
I = mv
I = (61kg)(29.7m/s)
I = 1818 kg m/s
To find the magnitude of the impulse exerted on the bungee jumper as the cord stretches, we can use the equation:
Impulse = Change in momentum
Step 1: Find the initial velocity of the bungee jumper
At the top of the bridge, the bungee jumper is momentarily at rest before falling. Thus, the initial velocity, v_0, is 0 m/s.
Step 2: Find the final velocity of the bungee jumper
To find the final velocity, v_f, we can use the equation of motion:
v_f^2 = v_0^2 + 2aΔy
where:
v_f is the final velocity (unknown)
v_0 is the initial velocity (0 m/s)
a is the acceleration (unknown)
Δy is the change in height (75 m - 5 m = 70 m)
Substituting the known values into the equation, we get:
v_f^2 = 0 + 2a(70 m)
Step 3: Find the acceleration
The acceleration can be calculated using the formula:
F_net = m * a
where:
F_net is the net force acting on the bungee jumper
m is the mass of the bungee jumper (61.00 kg)
At the maximum stretch of the cord, the net force acting on the bungee jumper is equal to the gravitational force acting on them:
F_net = (m * g) + (k * Δy)
where:
g is the acceleration due to gravity (9.8 m/s^2)
k is the spring constant of the cord (unknown)
Set this equation equal to m * a, and solve for a:
(m * g) + (k * Δy) = m * a
Since we are looking for the maximum stretch, the cord is at the equilibrium position (unstretched). Hence, the net force is zero at this point:
(0) + (k * Δy) = 0
Solve this equation for k:
k = 0 / Δy = 0
Substitute the value of k into the equation for acceleration:
(m * g) + (0 * Δy) = m * a
Rearrange the equation to solve for a:
a = (m * g) / m = g
The acceleration, a, is equal to the acceleration due to gravity: a = 9.8 m/s^2.
Step 4: Calculate the final velocity
Substituting the known values into the equation:
v_f^2 = 0 + 2(9.8 m/s^2)(70 m)
v_f^2 = 1372 m^2/s^2
Taking the square root of both sides gives us:
v_f = √1372 m/s
v_f ≈ 37.04 m/s (rounded to two decimal places)
Step 5: Calculate the change in momentum
The change in momentum is given by:
Δp = m(v_f - v_0)
Substituting the known values:
Δp = 61.00 kg(37.04 m/s - 0 m/s)
Δp ≈ 2262.44 kg*m/s (rounded to two decimal places)
The magnitude of the impulse exerted on the bungee jumper as the cord stretches is approximately 2262.44 kg*m/s.
To find the magnitude of the impulse exerted on the bungee jumper, we need to calculate the change in momentum. Here are the step-by-step calculations:
Step 1: Calculate the gravitational potential energy at the initial position:
Potential Energy = mass * gravity * height
Gravitational potential energy = 61.00 kg * 9.8 m/s^2 * 75.00 m
Step 2: Calculate the gravitational potential energy at the final position:
Gravitational potential energy = 61.00 kg * 9.8 m/s^2 * 5.00 m
Step 3: Calculate the change in gravitational potential energy:
Change in potential energy = Final potential energy - Initial potential energy
Step 4: Calculate the change in momentum:
Impulse = change in potential energy
To put these steps into action:
Step 1:
Gravitational potential energy = 61.00 kg * 9.8 m/s^2 * 75.00 m
Step 2:
Gravitational potential energy = 61.00 kg * 9.8 m/s^2 * 5.00 m
Step 3:
Change in potential energy = (61.00 kg * 9.8 m/s^2 * 5.00 m) - (61.00 kg * 9.8 m/s^2 * 75.00 m)
Step 4:
Impulse = Change in potential energy
By following these steps, you should be able to calculate the magnitude of the impulse exerted on the bungee jumper as the cord stretches.