2022 A/L Physics Past Paper · Question 13 (English Medium)

Answer: (1) Graph (1)

What the question asks

We want: the graph of $p$ against $\lambda$ for a moving particle.

What each letter means:

$p$ ($\text{kg m s}^{-1}$): the momentum of the particle.

$\lambda$ ($\text{m}$): the de Broglie wavelength of the particle.

$h$ ($\text{J s}$): the Planck constant, a fixed number.

Relevant theory

According to de Broglie's argument this relationship should hold for particle waves also.

This is known as the de Broglie wavelength of the particle.

For example, think of a bus trip of a fixed distance.

Speed times time always gives the same distance. So the faster the bus goes, the less time it takes.

Wavelength and momentum are tied in the same way. Their product is always one fixed number.

$$p\,\lambda = h$$

In words: momentum times wavelength is always the Planck constant. So more momentum means a shorter wavelength.

Required equations

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

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

Reasoning and calculation

Now draw $p = \dfrac{h}{\lambda}$.

Take a small $\lambda$. Then $p$ is very large.

Double $\lambda$. Then $p$ becomes half.

Double it again. Then $p$ becomes half again. It keeps getting smaller but never reaches zero.

Check the five graphs against this curve.

Graph (1) falls steeply at first and then flattens. It never touches the $\lambda$ axis. It matches.

Graph (2) is a rising line, graph (3) is a falling line that hits zero, graph (4) rises and graph (5) is flat.

None of these halves $p$ each time $\lambda$ doubles.

Checks

$l * p$

Does it make sense?

A fast, heavy ball has a large $p$. Its wavelength is so tiny that we never see it behave like a wave.

An electron has a small $p$. Its wavelength is big enough to measure. That is why electrons show wave effects.

Conclusion

$\text{graph }(1)$

$p\,\lambda = h$ is a fixed number, so $p$ falls as $\lambda$ grows and never reaches zero.

The answer is (1).

When two quantities multiply to a fixed number, one goes down as the other goes up, and the graph is a curve that never touches the axes.

Test the graph: double $\lambda$ and see if $p$ is halved.

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