Question

The velocity of proton is proportional to (where $v$ is frequency)
(A) $\dfrac{{{v}^{2}}}{2}$
(B) $\dfrac{1}{\sqrt{v}}$
(C) $\sqrt{v}$
(D) $v$

Answer 1

Write the relation between speed, distance and time. Now understand the term, frequency and its representation in terms of standard units. The formula given below will help you determine the relation between frequency and time.
$v=\dfrac{1}{t}$
Where,
$v$ is the frequency of the proton
$t$ is the time period

Complete step by step answer:
Velocity of an object can be defined as the rate of change of position of an object with respect to a frame of reference. It is a function of time.
It is a physical as well as a vector quantity. Thus, it requires both direction and magnitude in order to be defined.
Since velocity is the rate of change of position of an object, the relation between velocity, distance and time is given as,
$\text{v = }\dfrac{d}{t}$
Where,
$v$ is the velocity
$d$ is the distance
$t$ is the time taken
Frequency is the number of occurrences of a repeating event per unit time interval. At times it is also referred to as temporal frequency.
We will now use the relation between frequency and time given in the hint.
$v=\dfrac{1}{t}$
Substituting this in the equation of velocity, we get
$\text{v = d }\text{. }v$
Thus, the velocity of a proton is directly proportional to frequency.

Therefore, the correct answer is option (D).

Note: In case of a proton, we often take wavelength instead of mere distance as it propagates in a wave. Thus, the symbol d is often replaced by $\lambda$.