Children slide down a frictionless water slide that ends at a height of 1.80 m above the pool. If a child starts from rest at point A and lands in the water at point B, a horizontal distance L = 2.46 m from the base of the slide, determine the height h of the water slide.
Well, if there's no friction on the water slide, I hope there's also no clown trying to make you laugh while you slide down. That would be just slippingly funny! But let's get to the question.
To determine the height of the water slide, we can use the conservation of mechanical energy.
At point A, the child has potential energy due to being at a height h and no initial kinetic energy since they start from rest. At point B, the child has no potential energy because they are at the same level as the pool, but they have kinetic energy due to their horizontal velocity.
Using the principle of conservation of energy, we can say that the initial potential energy at point A is equal to the final kinetic energy at point B. Mathematically, this can be represented as:
mgh = (1/2)mv^2
Where m is the mass of the child, g is the acceleration due to gravity, h is the height of the water slide, and v is the horizontal velocity of the child.
Since the mass of the child cancels out, we can disregard it for this calculation.
Now, we know the horizontal distance L from the base of the slide to point B, so we can use some trigonometry to find the horizontal velocity v.
Since the child is sliding down a frictionless slide, we can say that the horizontal component of the child's velocity remains constant throughout the slide. This means that the child's velocity at point B is the same as their horizontal velocity at point A.
Using the equation:
sinθ = (opposite/hypotenuse)
We can write:
sinθ = L/h
Rearranging the equation to solve for h:
h = L/sinθ
So, if we know the value of L and the angle of the slide θ, we can calculate the height h of the water slide.
To find the height of the water slide, we can use the principle of conservation of energy. The energy at the top of the slide is converted into kinetic energy at the bottom of the slide.
Let's start by determining the potential energy at the top of the slide (point A) and the kinetic energy at the bottom of the slide (point B).
1. Potential energy at the top of the slide (point A):
Potential energy (PE) = mass (m) x gravity (g) x height (h)
2. Kinetic energy at the bottom of the slide (point B):
Kinetic energy (KE) = 1/2 x mass (m) x velocity squared (v^2)
Since the slide is frictionless, there is no energy loss due to friction, and all the potential energy is converted into kinetic energy.
Therefore, we can set up the equation:
PE = KE
mgh = 1/2 mv^2
Simplify the equation by canceling out mass (m):
gh = 1/2 v^2
Now, we need to determine the velocity of the child at the bottom of the slide (point B). We can use basic kinematic equations to find the velocity.
3. Using the kinematic equation:
v^2 = u^2 + 2as
Since the child starts from rest (u = 0) and there is no horizontal acceleration (a = 0), the equation becomes:
v^2 = 2as
Hence, v^2 = 2 x g x L (where L is the horizontal distance from the base of the slide to the point of landing)
Substitute this value of v^2 into equation (2):
gh = 1/2 x 2gL
Simplify:
gh = gL
Finally, isolate the height (h):
h = L
Therefore, the height of the water slide (h) is equal to the horizontal distance L, which is 2.46 m.
To determine the height h of the water slide, we can use the principle of conservation of energy.
Starting with the potential energy at point A and considering the kinetic energy at point B, we can equate the two:
Potential energy at A = Kinetic energy at B
The potential energy at point A can be calculated using the formula:
Potential energy = m * g * h
Where m is the mass of the child sliding down the slide, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height of the slide.
The kinetic energy at point B can be calculated using the formula:
Kinetic energy = (1/2) * m * v^2
Where v is the velocity of the child at point B. Since the child starts from rest, the initial velocity is 0, so the kinetic energy simplifies to:
Kinetic energy = 0
Now, setting the potential energy equal to the kinetic energy:
m * g * h = 0
Since the mass and acceleration due to gravity are constants, we can cancel them from both sides of the equation:
h = 0
Therefore, based on the given information, the height of the water slide is 0.
I assume A is at the top of the slide. This solution assumes the base of the slide releases the child in a horizontal direction.
time to fall to the water
1.80=1/2 g t
t=3.6/9.8 seconds
velocity at base of slide:
2.46=v*t=v*3.6/9.8
v= you solve that
now height of slide:
PE at top=KE at bottom
mgh=1/2 m v^2
h=1/2 v^2/g and you are done