a 40 kg child slides down a frictionless playground slide from a height of 2.0m.
What is her speed at the bottom of the slide?
Would the answer to part A change if she jumped straight down instead of sliding?
How would the answer to part A change if the slide were not frictionless?
Where does the lost energy go?
To find the speed of the child at the bottom of the slide, we can use the principle of conservation of mechanical energy. The initial energy of the child at the top of the slide is equal to the sum of her potential energy (mgh) and her initial kinetic energy (0, since she starts from rest). At the bottom of the slide, all her initial potential energy would be converted into kinetic energy.
Step-by-step calculation:
1. Calculate the potential energy (PE) at the top of the slide:
PE = mgh
where m is the mass (40 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the slide (2.0 m).
PE = (40 kg) x (9.8 m/s^2) x (2.0 m) = 784 J
2. The potential energy is converted into kinetic energy (KE) at the bottom of the slide:
KE = PE
KE = 784 J
3. Use the equation for kinetic energy to find the speed (v):
KE = (1/2)mv^2
where m is the mass (40 kg) and v is the velocity/speed.
Rearranging the equation to solve for v:
v = √(2KE/m)
v = √(2(784 J) / 40 kg)
v ≈ 7.0 m/s
Therefore, the speed of the child at the bottom of the slide is approximately 7.0 m/s.
Now, let's address the other questions:
Would the answer to part A change if she jumped straight down instead of sliding?
No, the answer to part A would not change if she jumped straight down instead of sliding. The speed at the bottom would still be approximately 7.0 m/s, assuming no air resistance.
How would the answer to part A change if the slide were not frictionless?
If the slide were not frictionless, there would be additional work done by friction, which would convert some of the child's potential energy into heat. This means that the kinetic energy at the bottom of the slide would be less than the potential energy at the top. Thus, the speed at the bottom of the slide would be lower than 7.0 m/s.
Where does the lost energy go?
The lost energy is primarily converted into heat due to friction between the child and the slide. This heat energy dissipates into the surrounding environment.
To determine the speed of the child at the bottom of the slide, we can use the principle of conservation of energy. The potential energy at the top of the slide is converted into kinetic energy at the bottom. Since there is no friction, there is no energy lost due to work done against it.
1. Calculate the potential energy at the top of the slide using the formula:
Potential Energy (PE) = mass (m) x gravity (g) x height (h)
PE = 40 kg x 9.8 m/s^2 x 2.0 m = 784 Joules
2. The potential energy is converted into kinetic energy at the bottom:
Kinetic Energy (KE) = 1/2 x mass (m) x velocity (v)^2
Set the potential energy equal to the kinetic energy and solve for velocity:
784 J = 1/2 x 40 kg x v^2
784 J = 20 kg x v^2
v^2 = 784 J / 20 kg
v^2 = 39.2 m^2/s^2
v = sqrt(39.2 m^2/s^2) = 6.26 m/s
Therefore, the child's speed at the bottom of the slide is 6.26 m/s.
If the child jumped straight down instead of sliding:
In this case, the potential energy at the top would still be the same, which is 784 Joules. However, now the total energy would be in the form of kinetic energy since there is no sliding or conversion of energy.
Using the formula for kinetic energy, we have:
Kinetic Energy (KE) = 1/2 x mass (m) x velocity (v)^2
Set the potential energy equal to the kinetic energy and solve for velocity:
784 J = 1/2 x 40 kg x v^2
784 J = 20 kg x v^2
v^2 = 784 J / 20 kg
v^2 = 39.2 m^2/s^2
v = sqrt(39.2 m^2/s^2) = 6.26 m/s
Therefore, the child's speed at the bottom would be the same, which is 6.26 m/s, regardless of whether they slide or jump straight down.
If the slide were not frictionless:
If the slide had some friction, the energy lost due to friction would be converted into heat. As a result, the amount of potential energy converted into kinetic energy would be less. This means the speed of the child at the bottom would be lower than 6.26 m/s. The exact change in speed would depend on the amount of friction present on the slide and the efficiency of the conversion of potential energy into kinetic energy.
Where does the lost energy go:
The energy lost due to friction on the slide is converted into heat. This could involve the heating of the slide surface or any other nearby objects that come into contact with the moving child.
look at energy:
KE at bottom=PEat top + KE initial0- fricton
if she started at zero velocity
1/2 m v^2=mg*2 + O - 0
solve for v at the bottom.
Now if she jumpted straight down, she would have KE initial, and v would be differnt.
If it had friction that woujld be in the equation above.
Lost energy: heat mainly