A 69.6 kg fire fighter slides down a pole while a constant frictional force of 263 N s his motion. A horizontal 18.7 kg platform is supported by a spring at the bottom of the pole to cushion the fall. The fire fighter starts from rest 5.80 m above the platform, and the spring constant is 2760 N/m. Calculate the fire fighter's speed just before he collides with the platform.
mgh-energylostslidingpole =final energy
mg(5.8)-263*5.8=1/2 m vf^2
solve for vf
I don't see any connection with the mass of the platform, or spring holding it up.
To calculate the firefighter's speed just before he collides with the platform, we can apply the principle of conservation of mechanical energy.
First, let's consider the gravitational potential energy of the firefighter at a height of 5.80 m above the platform:
PE_gravity = m * g * h
Where:
m = mass of the firefighter = 69.6 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height = 5.80 m
PE_gravity = 69.6 kg * 9.8 m/s^2 * 5.80 m
PE_gravity = 3782.16 J
Next, let's consider the elastic potential energy of the spring when the firefighter reaches the platform. The spring potential energy can be calculated using the formula:
PE_spring = 1/2 * k * x^2
Where:
k = spring constant = 2760 N/m
x = compression or displacement of the spring
Since the platform supports the firefighter when he reaches it, the displacement of the spring is equal to the height of the fall, in this case, 5.80 m.
PE_spring = 1/2 * 2760 N/m * (5.80 m)^2
PE_spring = 46526.40 J
Now, since there is friction acting against the motion of the firefighter as he slides down the pole, we need to take that into account as well. The work done by friction can be calculated as:
W_friction = F_friction * d
Where:
F_friction = frictional force = 263 N (given)
d = distance (or displacement) covered by the firefighter
Since the displacement covered by the firefighter is also the height he falls, we can use the distance of 5.80 m.
W_friction = 263 N * 5.80 m
W_friction = 1521.40 J
Now, let's consider the initial kinetic energy of the firefighter, which is zero since he starts from rest.
KE_initial = 0 J
Finally, since the total mechanical energy is conserved (assuming no other external forces), we can equate the initial mechanical energy to the final mechanical energy:
KE_initial + PE_gravity + PE_spring - W_friction = KE_final
Substituting the calculated values, we get:
0 + 3782.16 J + 46526.40 J - 1521.40 J = KE_final
44032.16 J = KE_final
Finally, we can use the equation for kinetic energy to solve for the final kinetic energy and then the speed:
KE_final = 1/2 * m * v^2
Where:
m = mass of the firefighter = 69.6 kg
v = speed
Rearranging the equation:
v^2 = (2 * KE_final) / m
v^2 = (2 * 44032.16 J) / 69.6 kg
v^2 = 1260 m^2/s^2
Taking the square root of both sides:
v = sqrt(1260 m^2/s^2)
v ≈ 35.5 m/s
Therefore, the fire fighter's speed just before he collides with the platform is approximately 35.5 m/s.