A 35-kg child starting from rest slides down a water slide with a vertical height of 20 m. What is the child’s speed a.)half-way down the slide’s vertical distance
b.) three-fourths the distance (ignoring friction)
a) Well, halfway down the slide, the child might realize they're too big to be going down a water slide and start questioning their life choices. But in terms of speed, we can calculate it using some physics. Since the child is sliding down with no friction, we can use the conservation of energy principle. The potential energy at the top is converted into kinetic energy at the bottom.
Using the formula for potential energy: PE = mgh (mass x gravity x height)
PE = (35 kg) x (9.8 m/s²) x (20 m) = 6860 Joules
Now, let's convert this potential energy into kinetic energy:
KE = (1/2)mv² (1/2 of mass x velocity squared)
6860 J = (1/2) x (35 kg) x v²
v² = (6860 J) x (2) / (35 kg)
v² ≈ 391.43 m²/s²
Finally, take the square root of both sides to find the speed:
v ≈ √(391.43 m²/s²)
v ≈ 19.79 m/s
So, halfway down the slide, the child's speed is approximately 19.79 m/s. Just make sure they don't forget to smile and wave!
b) Now, three-fourths of the distance down the slide, the child might start contemplating whether they can charge an entrance fee for their epic slide performance. Let's calculate their speed at this point!
Using the same principle of conservation of energy, we can determine the speed. The potential energy at this point is converted into kinetic energy.
Similar to the previous calculation, the potential energy at three-fourths the distance is:
PE = (35 kg) x (9.8 m/s²) x (20 m x 3/4)
PE = (35 kg) x (9.8 m/s²) x (15 m)
PE = 5145 Joules
Converting this potential energy into kinetic energy:
5145 J = (1/2) x (35 kg) x v²
v² = (5145 J) x (2) / (35 kg)
v² ≈ 294.43 m²/s²
Taking the square root of both sides:
v ≈ √(294.43 m²/s²)
v ≈ 17.14 m/s
So, three-fourths the distance down the slide, the child's speed is approximately 17.14 m/s. Make sure they don't go too fast and become a splashy spectacle at the end!
To determine the child's speed at different points on the slide, we can use the principles of conservation of energy.
a.) Half-way down the slide's vertical distance:
The potential energy at the highest point is converted to kinetic energy at half-way down the slide. Since there is no mention of friction, we can ignore it in this calculation.
Using the formula for potential energy (PE) and kinetic energy (KE):
PE = m * g * h
KE = 1/2 * m * v^2
Where:
m = mass of the child (35 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = height (20 m)
v = velocity
At the halfway point, the child has descended 10 m (half of the total height). Therefore, the potential energy (PE) at this point is:
PE = m * g * h
PE = 35 kg * 9.8 m/s^2 * 10 m
PE = 3430 J
This potential energy is completely converted to kinetic energy (KE):
KE = 1/2 * m * v^2
3430 J = 1/2 * 35 kg * v^2
v^2 = (3430 J) / (1/2 * 35 kg)
v^2 = 196 m^2/s^2
Taking the square root of both sides to solve for velocity (v):
v = √196 m^2/s^2
v = 14 m/s
Therefore, the child's speed at half-way down the slide's vertical distance is 14 m/s.
b.) Three-fourths the distance (ignoring friction):
Using the same principles, we can determine the child's speed at three-fourths of the distance down the slide.
At this point, the child has descended 15 m (three-fourths of the total height). Thus, the potential energy (PE) at this point is:
PE = m * g * h
PE = 35 kg * 9.8 m/s^2 * 15 m
PE = 5145 J
Again, this potential energy is completely converted to kinetic energy (KE):
KE = 1/2 * m * v^2
5145 J = 1/2 * 35 kg * v^2
v^2 = (5145 J) / (1/2 * 35 kg)
v^2 = 294 m^2/s^2
Taking the square root:
v = √294 m^2/s^2
v = 17.14 m/s
Therefore, the child's speed at three-fourths the distance down the slide (ignoring friction) is approximately 17.14 m/s.
To find the child's speed at different points along the water slide, we can apply the principle of conservation of energy. According to this principle, the potential energy at the top of the slide is converted into kinetic energy at different points down the slide.
The formula to calculate potential energy is given by:
Potential energy = mass × gravity × height
where:
mass = 35 kg (mass of the child)
gravity = 9.8 m/s^2 (acceleration due to gravity)
height = vertical distance traveled
a.) To find the child's speed halfway down the slide's vertical distance, we divide the height by 2:
height_halfway = 20 m / 2 = 10 m
The potential energy halfway down the slide is:
Potential energy_halfway = mass × gravity × height_halfway
= 35 kg × 9.8 m/s^2 × 10 m
Next, we can equate the potential energy at this point to the kinetic energy of the child:
Potential energy_halfway = Kinetic energy_halfway
Using the formula for kinetic energy:
Kinetic energy_halfway = (1/2) × mass × velocity^2
We can rearrange the equation to solve for the velocity:
(1/2) × mass × velocity^2 = Potential energy_halfway
mass × velocity^2 = 2 × Potential energy_halfway
velocity^2 = (2 × Potential energy_halfway) / mass
velocity = √((2 × Potential energy_halfway) / mass)
Now, plug in the values to calculate the velocity:
velocity_halfway = √((2 × Potential energy_halfway) / mass)
b.) To find the child's speed three-fourths the distance down the slide, we divide the height by 4 and multiply it by 3:
height_three_fourths = (20 m / 4) × 3 = 15 m
The potential energy three-fourths down the slide is:
Potential energy_three_fourths = mass × gravity × height_three_fourths
= 35 kg × 9.8 m/s^2 × 15 m
Using the same equation as above, we can calculate the velocity:
velocity_three_fourths = √((2 × Potential energy_three_fourths) / mass)
Now, you can plug in the values to find the child's speed at three-fourths the distance down the slide.
Use energy equation
mgh=(1/2)mv²
to solve for v:
v=√(2gh)
for h=5m, g=9.8m/s²
v=√(2*9.8*5)
=9.9 m/s in the direction along the slide.
You can solve for other heights in a similar way.