A girl is sitting on a sled. She is pushed by a spring that was compressed 0.8m. The spring constant is 2500 N/m. After moving 2.5 m across a surface that applies 60 n of friction, the girl and the sled slid up a frictionless hill. How high do the girl and the sled rise up by the time they stop?
spring energy = 1/2 k x^2 = 1/2 * 2500 * 0.8^2 ... Joules
friction work = f * d = 60 * 2.5 ... Joules
hill height = [(spring energy) - (friction work)] / (m * g)
... you need the girl/sled mass
the girl/sled mass is 27kg
Fs = 2500N/m * 0.8m = 2000 N.
Fs-Ff = M*a
2000-60 = 27a
a = 71.9 m/s^2.
V^2 = Vo^2 + 2a*d = 0 + 143.8*2.5 = 359.5
V = 19 m/s.
V^2 = Vo^2 + 2g*h = 0
19^2 + (-19.6)h = 0
h = 18.4 m.
To find how high the girl and the sled rise up the hill, we need to analyze the conservation of energy for the system.
First, let's consider the energy stored in the compressed spring. The potential energy stored in a spring can be calculated using the formula:
Potential energy (U) = 0.5 * k * (x^2)
Where:
k = spring constant
x = deformation or compression of the spring
Given:
Spring constant (k) = 2500 N/m
Compression of the spring (x) = 0.8 m
Substituting these values into the formula, we can find the potential energy stored in the spring.
U = 0.5 * 2500 N/m * (0.8 m)^2
U = 0.5 * 2500 N/m * 0.64 m^2
U = 800 N⋅m
Next, let's consider the work done against friction as the sled moves across the surface. The work done against friction can be calculated using the formula:
Work done against friction (W_friction) = force of friction * distance
Given:
Force of friction = 60 N
Distance = 2.5 m
Substituting these values into the formula, we can find the work done against friction.
W_friction = 60 N * 2.5 m
W_friction = 150 N⋅m
The work done against friction is converted into heat energy and does not contribute to the sled's upward motion. Therefore, we subtract the work done against friction from the potential energy stored in the spring to find the remaining energy available for lifting the sled up the hill.
Remaining energy = Potential energy - Work done against friction
Remaining energy = 800 N⋅m - 150 N⋅m
Remaining energy = 650 N⋅m
This remaining energy can be equated to the potential energy due to the vertical height gained by the girl and the sled. The potential energy can be calculated using the formula:
Potential energy (U) = m * g * h
Where:
m = mass of the girl and sled (assumed to be constant throughout the motion)
g = acceleration due to gravity (9.8 m/s^2)
h = height gained
Rearranging the formula to solve for height, we have:
h = U / (m * g)
Dividing the remaining energy by the product of mass and acceleration due to gravity, we can find the height gained by the sled.
Height (h) = 650 N⋅m / (m * 9.8 m/s^2)
Height (h) = 66.33 m / m
Unfortunately, without the mass of the girl and the sled, we can't determine the exact height they rise up.