A child with a mass of 45 kg is on top of a slide that is 12 m long and inclined 30 degrees along the horizontal. (a) Find the potential energy of the child at the top of the slide. How fast is the child going at the bottom of the slide if (b) the slide is frictionless and (c) if the coefficient of kinetic friction is 0.30?
Oh, sliding down the slide can be so much fun! Let me calculate the potential energy and the child's speed for you:
(a) To find the potential energy of the child at the top of the slide, we'll use the formula: potential energy = mass × gravity × height, where gravity is approximately 9.8 m/s^2.
Potential energy = 45 kg × 9.8 m/s^2 × 12 m × sin(30 degrees).
Calculating... (now the suspense builds)...
The potential energy of the child at the top of the slide is equal to approximately 3,137.1 joules. Keep holding on!
(b) Now, if the slide is frictionless, the potential energy will be completely converted into kinetic energy. So, using the conservation of energy, we can equate the potential energy to the kinetic energy at the bottom of the slide:
Potential energy = kinetic energy.
3,137.1 joules = (1/2) × mass × velocity^2.
We can solve for the child's velocity at the bottom of the slide:
Velocity = √(2 × potential energy / mass).
Plugging in the values...
The child's speed at the bottom of the frictionless slide is approximately 11.5 m/s. Wheeee!
(c) But what if there is friction on the slide? If the coefficient of kinetic friction is 0.30, we need to factor that in. Lengthening the ride, huh?
In this case, we need to consider the work done against friction, which reduces the kinetic energy at the bottom of the slide. We can calculate the work done by friction using the formula: work = force of friction × distance.
The force of friction can be calculated using the equation: force of friction = coefficient of kinetic friction × (mass × gravity).
And the work done against friction (distance) will be equal to the length of the slide, which is 12 m.
Using the work-energy theorem: work done = change in kinetic energy, we can calculate the change in kinetic energy:
Work done = change in kinetic energy = (1/2) × mass × (velocity^2 - 0).
Now, we'll calculate the velocity using the same equation as before:
Velocity = √(2 × (potential energy - work done) / mass).
Plugging in the values...
The child's speed at the bottom of the slide, with a coefficient of kinetic friction of 0.30, is approximately 9.2 m/s. It's a little slower due to friction, but still fun!
To solve this, we'll need to use the principles of mechanical energy.
(a) The potential energy of an object at a certain height can be calculated using the equation: 𝑃𝐸 = 𝑚𝑔ℎ, where 𝑚 is the mass of the object and ℎ is the height.
In this case, the height is the vertical distance from the top of the slide to the ground. Since the slide is inclined at an angle of 30 degrees, we can use trigonometry to find this height:
= 𝑙 × sin(𝜃)
= 12 m × sin(30°)
= 12 m × 0.5
= 6 m
Now we can calculate the potential energy:
𝑃𝐸 = 𝑚𝑔
𝑃𝐸 = 45 kg × 9.8 m/s^2 × 6 m
𝑃𝐸 = 2646 J
Therefore, the potential energy of the child at the top of the slide is 2646 Joules.
(b) If the slide is frictionless, the total mechanical energy is conserved. This means that the potential energy at the top of the slide will be converted into kinetic energy at the bottom, assuming no energy is lost due to friction.
The equation for kinetic energy is 𝐾𝐸 = 0.5𝑚𝑣^2, where 𝑣 is the velocity.
Since the potential energy is converted entirely into kinetic energy, we can equate the two:
𝑃𝐸 = 𝐾𝐸
2646 J = 0.5 × 45 kg × 𝑣^2
Simplifying the equation will give us:
𝑣^2 = 2646 J × 2 / 45 kg
𝑣^2 = 117.33 J/kg
Taking the square root, we find:
𝑣 = √117.33 J/kg
𝑣 ≈ 10.83 m/s
Therefore, if the slide is frictionless, the child will be going at a speed of approximately 10.83 m/s at the bottom of the slide.
(c) If the coefficient of kinetic friction is 0.30, we need to consider the energy lost due to friction. The work done by friction can be calculated using the equation: 𝐶𝑤 = 𝑚𝑔ℎ𝑓.
The frictional force 𝑓 can be calculated using the equation: 𝑓 = 𝑚𝑔𝜇𝑘, where 𝜇𝑘 is the coefficient of kinetic friction.
Substituting the values, we get:
𝑓 = 45 kg × 9.8 m/s^2 × 0.30
𝑓 ≈ 132.3 N
Now we can calculate the work done by friction:
𝐶𝑤 = 𝑓 × 𝑠, where 𝑠 is the distance along the slide (12 m)
𝐶𝑤 = 132.3 N × 12 m
𝐶𝑤 = 1587.6 J
This work done by friction represents the energy lost. Therefore, the kinetic energy at the bottom of the slide will be:
𝐾𝐸 = 𝑃𝐸 - 𝐶𝑤
𝐾𝐸 = 2646 J - 1587.6 J
𝐾𝐸 ≈ 1058.4 J
Now we can solve for the velocity using the equation for kinetic energy:
𝐾𝐸 = 0.5𝑚𝑣^2
1058.4 J = 0.5 × 45 kg × 𝑣^2
𝑣^2 = 1058.4 J × 2 / 45 kg
𝑣^2 ≈ 47.08 J/kg
Taking the square root, we find:
𝑣 ≈ √47.08 J/kg
𝑣 ≈ 6.86 m/s
Therefore, if the slide has a coefficient of kinetic friction of 0.30, the child will be going at a speed of approximately 6.86 m/s at the bottom of the slide.
To find the potential energy of the child at the top of the slide, we can use the formula:
Potential Energy = mass * acceleration due to gravity * height
First, we need to find the height of the slide. Since the slide is inclined at 30 degrees, we can use trigonometry to find the vertical height.
Height = length of the slide * sin(angle)
= 12 m * sin(30 degrees)
= 6 m * 0.5
= 3 m
Now, we can calculate the potential energy:
Potential Energy = 45 kg * 9.8 m/s^2 * 3 m
= 1323 J
So, the potential energy of the child at the top of the slide is 1323 Joules.
To find the child's speed at the bottom of the slide, we need to consider two scenarios: when the slide is frictionless and when there is kinetic friction.
(a) Frictionless Slide:
When the slide is frictionless, all the potential energy at the top of the slide is converted into kinetic energy at the bottom.
The formula for kinetic energy is:
Kinetic Energy = 0.5 * mass * speed^2
By equating the potential energy at the top to the kinetic energy at the bottom, we get:
Potential Energy = Kinetic Energy
m * g * h = 0.5 * m * v^2
Where:
m = mass of the child = 45 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height of the slide = 3 m
v = speed of the child at the bottom
Substituting the values into the equation:
45 kg * 9.8 m/s^2 * 3 m = 0.5 * 45 kg * v^2
Simplifying:
1323 J = 0.5 * 45 kg * v^2
v^2 = (2 * 1323 J) / 45 kg
v^2 = 58.84 m^2/s^2
v = √(58.84) m/s
v ≈ 7.68 m/s
So, the child's speed at the bottom of the frictionless slide is approximately 7.68 m/s.
(b) Slide with Coefficient of Kinetic Friction of 0.30:
When there is friction, some of the potential energy will be converted into heat due to the friction force.
The formula for kinetic energy in the presence of friction is:
Kinetic Energy = Work done by gravity - Work done by friction
Since the slide is inclined, the work done by gravity is given by:
Work done by gravity = m * g * h * cos(angle)
Substituting the values, we get:
Work done by gravity = 45 kg * 9.8 m/s^2 * 3 m * cos(30 degrees)
= 1323 J * 0.866 (as cos(30 degrees) = 0.866)
= 1146 J
To find the work done by friction, we can use the formula:
Work done by friction = friction force * distance
The friction force can be calculated using the formula:
Friction force = coefficient of friction * Normal force
The normal force is equal to the weight of the child, which is given by:
Normal force = m * g
Substituting the values:
Normal force = 45 kg * 9.8 m/s^2
= 441 N
Therefore, the friction force is:
Friction force = 0.30 * 441 N
= 132.3 N
Since the slide is 12 m long, the distance traveled is equal to the length of the slide.
Work done by friction = 132.3 N * 12 m
= 1587.6 J
Substituting both values into the formula for kinetic energy, we get:
Kinetic Energy = 1146 J - 1587.6 J
= -441.6 J
Since the result is negative, it means that all the potential energy is converted into heat due to friction, and the child will not have any kinetic energy left at the bottom of the slide.
So, when the coefficient of kinetic friction is 0.30, the child will not have any speed at the bottom of the slide.
sin30 = h/12
h = 6 m.
a. PE = mgh = 45*9.8*6 = ____Joules.
b. h = Vo*T+0.5gT^2 = 6
0+4.9T^2 = 6
T =