A pogo stick has a spring with a force constant of 2.85 ✕ 104 N/m, which can be compressed 11.5 cm. To what maximum height (in m) can a child jump on the stick using only the energy in the spring, if the child and stick have a total mass of 41.0 kg?
Well, let's bounce into this question, shall we? So, to find the maximum height the child can jump, we need to use the potential energy stored in the spring.
The potential energy stored in a spring is given by the equation U = (1/2) k x^2, where U is the potential energy, k is the force constant, and x is the displacement of the spring.
In this case, the potential energy is going to equal the maximum height the child can reach, and the displacement, x, is going to be the compression of the spring, which is 11.5 cm (or 0.115 m).
So, plugging in the values into the equation, we have:
U = (1/2) (2.85 ✕ 104 N/m) (0.115 m)^2
Now, let's crunch some numbers, shall we?
U = 194.59 J
So, the maximum height the child can reach is 194.59 J. Now, it's important to keep in mind that the potential energy gets converted into kinetic energy as the child jumps, so there might be some loss of energy to consider as well.
But hey, let's not turn this into a circus act! To answer your question, the maximum height the child can reach on the pogo stick is approximately 194.59 meters. Happy bouncing!
To determine the maximum height a child can jump on the pogo stick using only the energy in the spring, we need to consider the conservation of mechanical energy.
Step 1: Calculate the potential energy stored in the compressed spring.
The potential energy stored in a compressed spring can be calculated using the formula:
PE = (1/2) * k * x^2
Where PE is the potential energy, k is the spring constant, and x is the compression of the spring.
Given:
k = 2.85 x 10^4 N/m
x = 11.5 cm = 0.115 m
PE = (1/2) * 2.85 x 10^4 N/m * (0.115 m)^2
Step 2: Calculate the maximum height using the potential energy.
The potential energy stored in the spring will be converted to gravitational potential energy when the child jumps on the pogo stick.
The formula for gravitational potential energy is:
PE = m * g * h
Where m is the mass of the child and the pogo stick, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the maximum height.
Given:
m = 41.0 kg
g = 9.8 m/s^2
Set the potential energy stored in the spring equal to the gravitational potential energy:
(1/2) * 2.85 x 10^4 N/m * (0.115 m)^2 = 41.0 kg * 9.8 m/s^2 * h
Step 3: Solve for the maximum height (h).
First, simplify the equation:
1.6475 x 10^2 N * m = 401.8 kg * m/s^2 * h
Divide both sides by 401.8 kg * m/s^2:
h = (1.6475 x 10^2 N * m) / (401.8 kg * m/s^2)
h ≈ 0.410 m
Therefore, the child can jump to a maximum height of approximately 0.410 meters using only the energy in the spring.
To find the maximum height the child can jump using only the energy in the spring, we need to calculate the potential energy stored in the spring when it is compressed and then equate it to the potential energy gained by the child when jumping.
1. Calculate the potential energy stored in the spring:
The potential energy stored in a spring can be calculated using the formula:
Potential Energy = 0.5 * k * x^2
where k is the force constant of the spring and x is the displacement/compression of the spring.
In this case, k = 2.85 * 10^4 N/m and x = 11.5 cm = 0.115 m.
Potential Energy = 0.5 * 2.85 * 10^4 N/m * (0.115 m)^2
2. Calculate the potential energy gained by the child:
The potential energy gained by the child is given by the formula:
Potential Energy = m * g * h
where m is the mass of the child and stick (41.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the maximum height reached.
Equate the potential energy stored in the spring to the potential energy gained by the child:
0.5 * 2.85 * 10^4 N/m * (0.115 m)^2 = 41.0 kg * 9.8 m/s^2 * h
3. Solve for h (maximum height):
Rearrange the equation and solve for h:
h = (0.5 * 2.85 * 10^4 N/m * (0.115 m)^2) / (41.0 kg * 9.8 m/s^2)
Calculate h using the given values.
Note: Make sure to convert the given values into consistent units (e.g., meters).
By evaluating the calculations, the maximum height the child can jump on the pogo stick using only the energy in the spring is approximately 0.31 m.
1/2 k x^2 = m g h
1/2 * 2.85E4 * .115^2 = 41.0 * 9.81 * h