A 30.0kg child standing from rest slides down a water slide with a vertical height of 10.0m. What is the childs speed (a)halfway down he slides vertical distance and (b)three-fourths of the way down? (Neglect friction?

ΔPE=KE

mgΔh=mv²/2
v=sqrt(2gΔh)
(a) Δh=5 m,
(b) Δh=7.5 m.

34.5

To find the child's speed halfway down the slide, we can use the principle of conservation of energy. The total initial energy of the child is equal to the total final energy at halfway down the slide.

Step 1: Calculate the potential energy at the top of the slide (P.E.):
Potential energy (P.E.) = mass (m) * acceleration due to gravity (g) * height (h)
P.E. = 30.0 kg * 9.8 m/s^2 * 10.0 m
P.E. = 2940 J

Step 2: Calculate the kinetic energy halfway down the slide (K.E.):
At halfway down the slide, all the potential energy would have been converted to kinetic energy.
K.E. = P.E. at the top of the slide
K.E. = 2940 J

Step 3: Calculate the speed halfway down the slide (v):
Kinetic energy (K.E.) = (1/2) * mass (m) * velocity (v)^2
2940 J = (1/2) * 30.0 kg * v^2
v^2 = (2940 J * 2) / (30.0 kg)
v^2 = 196 m^2/s^2
v = √(196 m^2/s^2)
v = 14 m/s

Therefore, the child's speed halfway down the slide is 14 m/s.

To find the child's speed three-fourths of the way down the slide, we'll repeat the same steps, but this time considering the change in height.

Step 1: Calculate the potential energy at three-fourths of the slide's height (P.E.):
Potential energy (P.E.) = mass (m) * acceleration due to gravity (g) * height (h)
P.E. = 30.0 kg * 9.8 m/s^2 * (10.0 m * 3/4)
P.E. = 2205 J

Step 2: Calculate the kinetic energy three-fourths of the way down the slide (K.E.):
At three-fourths of the way down the slide, the potential energy would have been converted to kinetic energy.
K.E. = P.E. at three-fourths of the slide's height
K.E. = 2205 J

Step 3: Calculate the speed three-fourths of the way down the slide (v):
Kinetic energy (K.E.) = (1/2) * mass (m) * velocity (v)^2
2205 J = (1/2) * 30.0 kg * v^2
v^2 = (2205 J * 2) / (30.0 kg)
v^2 = 147 m^2/s^2
v = √(147 m^2/s^2)
v = 12.12 m/s

Therefore, the child's speed three-fourths of the way down the slide is approximately 12.12 m/s.

To find the child's speed halfway down the slide and three-fourths of the way down, we need to use the principles of conservation of energy. The potential energy of the child at the top of the slide is converted into kinetic energy as the child slides down.

Let's first calculate the potential energy of the child at the top of the slide:

Potential energy (PE) = mass (m) * gravity (g) * height (h)

Given:
Mass (m) = 30.0 kg
Gravity (g) = 9.8 m/s^2
Height (h) = 10.0 m

Potential energy at the top of the slide:
PE = 30.0 kg * 9.8 m/s^2 * 10.0 m = 2940 Joules

Now, at halfway down, half of the potential energy is converted into kinetic energy, and at three-fourths of the way down, three-fourths of the potential energy is converted into kinetic energy.

Let's calculate the speed at halfway down the slide:

Kinetic energy (KE) = (1/2) * mass (m) * velocity^2

Since the potential energy is converted entirely into kinetic energy at halfway down, we can equate the potential energy at the top to the kinetic energy at halfway down:

Potential energy at the top = Kinetic energy at halfway down

Thus,

PE = KE
2940 Joules = (1/2) * 30.0 kg * velocity^2

Simplifying,

velocity^2 = (2940 Joules * 2) / 30.0 kg
velocity^2 = 196 m^2/s^2
velocity = √(196 m^2/s^2)
velocity = 14 m/s

Therefore, the child's speed halfway down the slide is 14 m/s.

Now, let's calculate the speed at three-fourths of the way down the slide:

Since three-fourths of the potential energy is converted into kinetic energy at three-fourths of the way down, we can calculate the remaining potential energy and equate it to the kinetic energy at that point.

Potential energy remaining = 1/4 * potential energy at the top

Potential energy remaining = 1/4 * 2940 Joules
Potential energy remaining = 735 Joules

Now, let's equate the potential energy remaining to the kinetic energy at three-fourths of the way down:

Potential energy remaining = Kinetic energy at three-fourths down

735 Joules = (1/2) * 30.0 kg * velocity^2

Simplifying,

velocity^2 = (735 Joules * 2) / 30.0 kg
velocity^2 = 49 m^2/s^2
velocity = √(49 m^2/s^2)
velocity = 7 m/s

Therefore, the child's speed three-fourths of the way down the slide is 7 m/s.