A 30.0 kg child slides down a long slide in a playground. She starts from rest at a height h1 of 18.00 m. When she is partway down the slide, at a height h2 of 9.00 m, she is moving at a speed of 9.30 m/s. Calculate the mechanical energy lost due to friction (as heat, etc.).
loss of potential energy = m g (18-9)
so
mg(18-9) = (1/2) m v^2 + loss due to friction
or
30(9.8)(9) = 15(9.3)^2 + loss due to friction
Why did the child bring a slide rule to the playground? To calculate the "slippery" slope of fun, of course!
Alright, let's slide into this physics problem. We can start by determining the child's initial gravitational potential energy at height h1. Using the formula U(mgh), where m is the mass of the child (30.0 kg) and g is the acceleration due to gravity (approximately 9.8 m/s²), we find:
U1 = (30.0 kg) * (9.8 m/s²) * (18.00 m) = 5,294.4 J
Next, we can calculate the child's final gravitational potential energy at height h2. Using the same formula, but with h2 as the new height and the same values for mass and gravitational acceleration, we get:
U2 = (30.0 kg) * (9.8 m/s²) * (9.00 m) = 2,646.0 J
The mechanical energy lost due to friction can be calculated as the difference between these two gravitational potential energies:
Mechanical energy lost = U1 - U2 = 5,294.4 J - 2,646.0 J = 2,648.4 J
So, the mechanical energy lost due to friction is 2,648.4 Joules. But hey, at least the child gained a thrilling ride and a few giggles along the way!
To calculate the mechanical energy lost due to friction, we first need to find the initial mechanical energy and the final mechanical energy of the child.
The initial mechanical energy (Ei) is given by the gravitational potential energy at height h1, and the final mechanical energy (Ef) is given by the sum of the child's kinetic energy and gravitational potential energy at height h2.
The formula for gravitational potential energy (PE) is:
PE = m * g * h
Where:
m = mass of the child = 30.0 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height
Let's calculate the initial and final mechanical energy:
Initial mechanical energy (Ei):
Ei = PE1 = m * g * h1
Ei = 30.0 kg * 9.8 m/s^2 * 18.00 m
Ei = 5292 J
Final mechanical energy (Ef):
Ef = PE2 + KE2
PE2 = m * g * h2
KE2 = 0.5 * m * v^2
PE2 = 30.0 kg * 9.8 m/s^2 * 9.00 m
PE2 = 2646 J
KE2 = 0.5 * 30 kg * (9.30 m/s)^2
KE2 = 1301.85 J
Ef = PE2 + KE2
Ef = 2646 J + 1301.85 J
Ef = 3947 J
To find the mechanical energy lost due to friction, we subtract the final mechanical energy (Ef) from the initial mechanical energy (Ei):
Mechanical energy lost = Ei - Ef
Mechanical energy lost = 5292 J - 3947 J
Mechanical energy lost = 1345 J
Therefore, the mechanical energy lost due to friction is 1345 J.
To calculate the mechanical energy lost due to friction, we need to find the initial mechanical energy and the final mechanical energy. Then, the difference between them will give us the mechanical energy lost.
1. Calculate the initial mechanical energy (E1):
The initial mechanical energy of the child is the sum of her potential energy and kinetic energy at the starting point (h1 = 18.00 m).
Potential energy at h1:
PE1 = m * g * h1
where m is the mass of the child (30.0 kg) and g is the acceleration due to gravity (9.8 m/s^2).
Kinetic energy at h1 (since she starts from rest):
KE1 = 0
Total initial mechanical energy (E1):
E1 = PE1 + KE1
2. Calculate the final mechanical energy (E2):
The final mechanical energy of the child is the sum of her potential energy and kinetic energy at the midway point (h2 = 9.00 m).
Potential energy at h2:
PE2 = m * g * h2
Kinetic energy at h2:
KE2 = 0.5 * m * v^2
where v is the velocity of the child at h2 (9.30 m/s).
Total final mechanical energy (E2):
E2 = PE2 + KE2
3. Calculate the mechanical energy lost (ΔE):
Mechanical energy lost due to friction is given by the difference between the initial and final mechanical energies.
ΔE = E1 - E2
Now, substitute the given values into the equations and solve for the mechanical energy lost due to friction.