Chapter 27, Problem 45P

To determine

The speed at which car must be moving to diffract and how diffraction conditions compare when the speed of the car is $30\text{m/s}$.

Answer to Problem 45P

Solution:

• The speed of the car must be $3.94\times {10}^{-38}\text{m/s}$.

• When the speed of the car is $30\text{m/s}$, there will be no diffraction.

Explanation of Solution

Given:

Mass of the car: $1400\text{kg}$

Length of the underpass: $12\text{m}$

Formula Used:

$\lambda =\frac{h}{p}$

Calculation:

⮚ For diffraction to occur the wavelength must be of the order of slit or opening. For the car to be diffracting the wavelength must be of the order of the opening. So $\lambda =12\text{m}$.

Now, by using the formula $\lambda =\frac{h}{p}$, the speed of the car can be calculated

$\begin{array}{c}\lambda =\frac{h}{p}=\frac{h}{mv}\\ v=\frac{h}{m\lambda }\\ v=\frac{(6.63\times {10}^{-34}\text{J}\text{.s})}{(1400\text{kg})(12\text{m})}\\ v=3.94\times {10}^{-38}\text{m/s}\end{array}$

For the car to be diffract, the speed of the car must be $3.94\times {10}^{-38}\text{m/s}$.

⮚ When the speed of the car is $30\text{m/s}$, the wavelength of the car will be

$\begin{array}{l}\lambda =\frac{h}{mv}\\ \lambda =\frac{(6.63\times {10}^{-34}\text{J}\text{.s})}{(1400\text{kg})(30\text{m/s})}\\ \lambda =1.57\times {10}^{-36}\text{m}\end{array}$

We see that the value of wavelength is much shorter than the opening i.e. 12 m. so there will be no diffraction