Worksheet frequency wavelength Energy - Speed /Frequency ... - Free Printable

Let's solve each problem step by step using the fundamental wave equation:

Wave Equation:

$$
\text{Speed} = \text{Frequency} \times \text{Wavelength}
$$

For electromagnetic waves (like light, radio waves, X-rays), the speed is the speed of light:
$$
c = 3.0 \times 10^8 \, \text{m/s}
$$

So:
- $ f = \frac{c}{\lambda} $
- $ \lambda = \frac{c}{f} $

---

Problem 1: Violet Light

Given:
Wavelength $ \lambda = 4.10 \times 10^{-12} \, \text{m} $
Find frequency $ f $

$$
f = \frac{c}{\lambda} = \frac{3.0 \times 10^8}{4.10 \times 10^{-12}} = \frac{3.0}{4.10} \times 10^{8 + 12} = 0.7317 \times 10^{20} = 7.32 \times 10^{19} \, \text{Hz}
$$

✔ Answer: $ 7.32 \times 10^{19} \, \text{Hz} $

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Problem 2: Green Light

Given:
Frequency $ f = 6.01 \times 10^{14} \, \text{Hz} $
Find wavelength $ \lambda $

$$
\lambda = \frac{c}{f} = \frac{3.0 \times 10^8}{6.01 \times 10^{14}} = \frac{3.0}{6.01} \times 10^{8 - 14} = 0.499 \times 10^{-6} = 4.99 \times 10^{-7} \, \text{m}
$$

✔ Answer: $ 4.99 \times 10^{-7} \, \text{m} $ or 499 nm

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Problem 3: Radio Station (640 kHz)

Given:
Frequency $ f = 640 \, \text{kHz} = 640 \times 10^3 \, \text{Hz} = 6.40 \times 10^5 \, \text{Hz} $

$$
\lambda = \frac{3.0 \times 10^8}{6.40 \times 10^5} = \frac{3.0}{6.40} \times 10^{8 - 5} = 0.46875 \times 10^3 = 468.75 \, \text{m}
$$

✔ Answer: $ 469 \, \text{m} $ (approximately)

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Problem 4: Radiation with $ f = 8.0 \times 10^{14} \, \text{Hz} $

$$
\lambda = \frac{3.0 \times 10^8}{8.0 \times 10^{14}} = \frac{3.0}{8.0} \times 10^{-6} = 0.375 \times 10^{-6} = 3.75 \times 10^{-7} \, \text{m}
$$

✔ Answer: $ 3.75 \times 10^{-7} \, \text{m} $ or 375 nm

---

Problem 5: Light with $ f = 7.66 \times 10^{14} \, \text{Hz} $

$$
\lambda = \frac{3.0 \times 10^8}{7.66 \times 10^{14}} = \frac{3.0}{7.66} \times 10^{-6} \approx 0.3917 \times 10^{-6} = 3.92 \times 10^{-7} \, \text{m}
$$

✔ Answer: $ 3.92 \times 10^{-7} \, \text{m} $ or 392 nm

---

Problem 6: Helium Laser, $ \lambda = 633 \, \text{nm} $

Convert to meters:
$ 633 \, \text{nm} = 633 \times 10^{-9} \, \text{m} = 6.33 \times 10^{-7} \, \text{m} $

$$
f = \frac{3.0 \times 10^8}{6.33 \times 10^{-7}} = \frac{3.0}{6.33} \times 10^{8 + 7} = 0.474 \times 10^{15} = 4.74 \times 10^{14} \, \text{Hz}
$$

✔ Answer: $ 4.74 \times 10^{14} \, \text{Hz} $

---

Problem 7: X-rays, $ f = 4.80 \times 10^{17} \, \text{Hz} $

$$
\lambda = \frac{3.0 \times 10^8}{4.80 \times 10^{17}} = \frac{3.0}{4.80} \times 10^{-9} = 0.625 \times 10^{-9} = 6.25 \times 10^{-10} \, \text{m}
$$

✔ Answer: $ 6.25 \times 10^{-10} \, \text{m} $

---

Problem 8: FM Radio Station, $ f = 107.9 \, \text{MHz} $

Convert MHz → Hz:
$ 107.9 \, \text{MHz} = 107.9 \times 10^6 \, \text{Hz} = 1.079 \times 10^8 \, \text{Hz} $

$$
\lambda = \frac{3.0 \times 10^8}{1.079 \times 10^8} = \frac{3.0}{1.079} \approx 2.78 \, \text{m}
$$

✔ Answer: $ 2.78 \, \text{m} $

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Problem 9: Sound Wavelengths (Human Hearing Limits)

Use sound speed: $ v = 345 \, \text{m/s} $

#### a) Frequency = 20 Hz
$$
\lambda = \frac{v}{f} = \frac{345}{20} = 17.25 \, \text{m}
$$

✔ Answer: $ 17.25 \, \text{m} $

#### b) Frequency = 20,000 Hz
$$
\lambda = \frac{345}{20,000} = 0.01725 \, \text{m} = 1.725 \, \text{cm}
$$

✔ Answer: $ 0.01725 \, \text{m} $ or 1.725 cm

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $ 7.32 \times 10^{19} \, \text{Hz} $ |
| 2 | $ 4.99 \times 10^{-7} \, \text{m} $ |
| 3 | $ 469 \, \text{m} $ |
| 4 | $ 3.75 \times 10^{-7} \, \text{m} $ |
| 5 | $ 3.92 \times 10^{-7} \, \text{m} $ |
| 6 | $ 4.74 \times 10^{14} \, \text{Hz} $ |
| 7 | $ 6.25 \times 10^{-10} \, \text{m} $ |
| 8 | $ 2.78 \, \text{m} $ |
| 9a | $ 17.25 \, \text{m} $ |
| 9b | $ 0.01725 \, \text{m} $ |

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