From a cliff that is 8.9 m above a lake, a young woman (mass 44 kg) jumps from rest, straight down into the water. At the instant she strikes the water, what is her de Broglie wavelength?
To find the de Broglie wavelength of the young woman, we need to use the equation:
λ = h / p
Where:
λ = de Broglie wavelength
h = Planck's constant (6.626 × 10^-34 J·s)
p = momentum
The momentum (p) of an object can be calculated as the product of its mass (m) and velocity (v):
p = m * v
In this case, the mass of the young woman is given as 44 kg. Now we need to find her velocity when she strikes the water.
To do that, we can apply the principle of conservation of energy. The potential energy that the woman initially has due to her height above the lake will be converted into kinetic energy just before she hits the water.
The potential energy (PE) can be calculated using the formula:
PE = m * g * h
Where:
m = mass (44 kg)
g = acceleration due to gravity (9.8 m/s²)
h = height (8.9 m)
Now, let's solve for the potential energy:
PE = 44 kg * 9.8 m/s² * 8.9 m
PE ≈ 3,985 J
This potential energy will be converted into kinetic energy just before she hits the water:
KE = PE
Now we can use the equation for kinetic energy:
KE = (1/2) * m * v²
Substituting the values:
3,985 J = (1/2) * 44 kg * v²
Now, let's solve for the velocity (v):
7,970 J = 22 kg * v²
v² = 362.273 m²/s²
v ≈ 19.03 m/s
Finally, we can calculate the de Broglie wavelength:
λ = h / p
λ = 6.626 × 10^-34 J·s / (44 kg * 19.03 m/s)
Now we can solve for the de Broglie wavelength of the young woman.