An Olympic diver performs a dive from the 10 m platform and achieves a speed of 14 m/s just before entering the water. Neglecting air friction, approximately how many times larger the diver’s speed than the diver would have achieved by performing from a 1 m platform? How much more potential energy does the diver have from the 10 m platform than on a 1 m platform?

To determine how many times larger the diver's speed is from the 10 m platform compared to the 1 m platform, we can use the principle of conservation of energy. The initial potential energy from which the diver falls is converted into kinetic energy just before entering the water. Since the diver's mass remains the same, we only need to consider the difference in height.

The potential energy of an object at a height h is given by the formula: PE = mgh

Let's first calculate the potential energy at each platform height.

For the 10 m platform:
PE10 = mg * h10

And for the 1 m platform:
PE1 = mg * h1

Now, the potential energy difference between the two platforms is given by the equation:

ΔPE = PE10 - PE1

Since the mass and gravity are constant, we can simplify the equation to:

ΔPE = mg * (h10 - h1)

To determine the ratio of the diver's speed from the 10 m platform to the 1 m platform, we need to compare their kinetic energies just before entering the water. The formula for kinetic energy is: KE = 1/2 * m * v^2

Let's calculate the kinetic energy from each platform height.

For the 10 m platform:
KE10 = 1/2 * m * v10^2

And for the 1 m platform:
KE1 = 1/2 * m * v1^2

The ratio can then be calculated by dividing the kinetic energy from the 10 m platform by the kinetic energy from the 1 m platform:

Ratio = KE10 / KE1 = (1/2 * m * v10^2) / (1/2 * m * v1^2)

Since the mass cancels out, we are left with:

Ratio = (v10^2) / (v1^2)

Now, substitute the provided values:

v10 = 14 m/s
v1 = ?

Solving for v1, we rearrange the formula:

14^2 = v1^2

Taking the square root of both sides gives us:

v1 = √(14^2) = 14

Therefore, the diver would achieve a speed of 14 m/s from the 1 m platform.

Now, substitute the values into the formula to determine the ratio:

Ratio = (14^2) / (14^2)
Ratio = 1

This means that the diver's speed from the 10 m platform is the same as the diver's speed from the 1 m platform.

To calculate the potential energy difference, substitute the given values:

h10 = 10 m
h1 = 1 m

ΔPE = mg * (h10 - h1)
ΔPE = m * g * (10 - 1)
ΔPE = m * g * 9

Here, g is the acceleration due to gravity, which is approximately 9.8 m/s^2. So, substitute and calculate:

ΔPE = m * (9.8) * 9
ΔPE = 88.2 m * g

Therefore, the potential energy difference between the 10 m platform and the 1 m platform is 88.2 times the product of the diver's mass and the acceleration due to gravity.