how fast is an electron moving if it has a wavelength equal to the distance travelled in one second ?

Options

A. `3.5 xx 10^(-8) m`
B. `sqrt((m)/(h))`
C. `sqrt((h)/(m))`
D. `sqrt((h)/(2K.E))`

Answer

To solve the problem of how fast an electron is moving if it has a wavelength equal to the distance traveled in one second, we can follow these steps:

### Step 1: Understand the relationship between wavelength and velocity
According to the problem, the wavelength (λ) of the electron is equal to the distance it travels in one second. The distance traveled in one second by an object is defined as its velocity (v). Therefore, we can write:
\[ \lambda = v \]

### Step 2: Use the de Broglie wavelength formula
The de Broglie wavelength of a moving particle is given by the formula:
\[ \lambda = \frac{h}{mv} \]
where:
- \( h \) is the Planck constant,
- \( m \) is the mass of the particle,
- \( v \) is the velocity of the particle.

### Step 3: Substitute the relationship from Step 1 into the de Broglie equation
Since we have established that \( \lambda = v \), we can substitute this into the de Broglie equation:
\[ v = \frac{h}{mv} \]

### Step 4: Rearrange the equation to solve for velocity
To isolate \( v \), we can multiply both sides by \( mv \):
\[ v^2 = \frac{h}{m} \]
Now, we can take the square root of both sides to find \( v \):
\[ v = \sqrt{\frac{h}{m}} \]

### Step 5: Conclusion
The speed of the electron is given by:
\[ v = \sqrt{\frac{h}{m}} \]