a) A sprinter must average 24.0 mi/h to win a 100-m dash in 9.30 s. What is his wavelength at this speed if his mass is 84.5 kg?
b) For potassium metal, the work function W (the minimum energy needed to eject an electron from the metal surface) is 3.68 × 10-19 J. Which is the longest wavelength of the following which could excite photoelectrons?
(550 nm, 500 nm, 450 nm, 400 nm, 350 nm)
The easiest way to do 2 is to calculate the energy of each wavelength listed.
For #2, wavelength = h/mv
Of course you must convert 24.0 mi/h to m/s. The easy way to do that is to type in google "24.9 miles/hour to m/s" without the parentheses and hit the enter key.
550
a) To find the wavelength of the sprinter, we can use the formula:
wavelength = speed / frequency
Since we are given the speed (24.0 mi/h), we need to convert it to meters per second (m/s):
24.0 mi/h * 1609.34 m/mi / 3600 s/h = 10.741 m/s (rounded to three decimal places)
Next, we need to find the frequency. The frequency can be calculated using the formula:
frequency = 1 / time
The time given is 9.30 s, so the frequency is:
frequency = 1 / 9.30 s = 0.1075 Hz (rounded to four decimal places)
Now we can calculate the wavelength:
wavelength = speed / frequency = 10.741 m/s / 0.1075 Hz = 99.999 meters (rounded to three decimal places)
Therefore, the sprinter's wavelength at this speed is approximately 99.999 meters.
b) The longest wavelength of light that can excite photoelectrons is given by the equation:
wavelength = speed of light / frequency
We know the speed of light is approximately 3.00 × 10^8 m/s. To find the frequency, we can use the equation:
frequency = energy / Planck's constant
The longest wavelength corresponds to the lowest energy, which is the work function, W= 3.68 × 10^-19 J. Planck's constant is given by h = 6.626 × 10^-34 J·s.
Now we can calculate the frequency:
frequency = energy / Planck's constant = (3.68 × 10^-19 J) / (6.626 × 10^-34 J·s) = 5.558 × 10^14 Hz (rounded to three decimal places)
Finally, we can calculate the longest wavelength:
wavelength = speed of light / frequency = (3.00 × 10^8 m/s) / (5.558 × 10^14 Hz) = 539.75 nm (rounded to two decimal places)
Therefore, the longest wavelength that could excite photoelectrons is approximately 539.75 nm, which is closest to 550 nm from the given options.
a) To find the wavelength of the sprinter at a speed of 24.0 mi/h, we need to first convert the speed to meters per second (m/s) because wavelength is usually expressed in meters.
1 mile = 1609.34 meters (approximately)
24.0 mi/h = (24.0 * 1609.34) m / (1 h * 3600 s) = 10.67 m/s (approximately)
Next, we need to determine the frequency of the sprinter. Since speed is the product of wavelength and frequency (speed = wavelength * frequency), we can rearrange the formula to solve for wavelength (wavelength = speed / frequency).
We are given that the sprinter completes the 100-meter dash in 9.30 seconds, which gives us the time taken (9.30 s). The frequency is the reciprocal of the time (frequency = 1 / time).
Therefore, frequency = 1 / 9.30 s = 0.1075 Hz
Now we can calculate the wavelength by dividing the speed by the frequency:
wavelength = 10.67 m/s / 0.1075 Hz = 99.19 meters (approximately)
So, at a speed of 24.0 mi/h, the wavelength of the sprinter is approximately 99.19 meters.
b) To determine the longest wavelength that can excite photoelectrons for potassium metal, we need to use the formula relating energy (E) and wavelength (λ), which is E = hc / λ, where h is Planck's constant (6.626 x 10^-34 J·s) and c is the speed of light (3.00 x 10^8 m/s).
We are given the work function (W) for potassium metal, which is the minimum energy required to eject an electron from the metal surface. The energy of a photon needs to be greater than or equal to the work function in order to excite photoelectrons.
Using the formula E = hc / λ, we can rearrange it to solve for λ:
λ = hc / E
Now we can substitute the given values:
λ = (6.626 x 10^-34 J·s * 3.00 x 10^8 m/s) / (3.68 x 10^-19 J)
Calculating this expression gives us:
λ = 1.70 x 10^-7 meters (approximately)
Since the answer choices are given in nanometers (nm), we need to convert the wavelength from meters to nanometers. There are 1 billion nanometers in a meter, so:
λ = 1.70 x 10^-7 meters * 1 x 10^9 nm / 1 meter
This simplifies to:
λ = 170 nm
Therefore, the longest wavelength that could excite photoelectrons for potassium metal is 170 nm.