Let two bodies be with masses 2kg and 5kg respectively. Let these bodies be at rest with the same force acting on them. Calculate the ratio of times that is required by both the bodies to reach the final velocity

The correct option is "2:5".

To understand why this is the correct answer, let us break down the physical principles and equations step-by-step.

1. Understand the Given Information:
Let the mass of the first body be $$m₁ = 2 kg.
Let the mass of the second body be $m$₂ = 5 kg.
Both bodies start from rest, which means their initial velocity ($u$) is zero:
$u$₁ = u₂ = 0.
The same constant force ($F$) acts on both bodies:
$F$₁ = F₂ = F.
We need to find the ratio of the times ($t$₁ : t₂) required by both bodies to reach the same final velocity ($v$), so:
$v$₁ = v₂ = v.

2. Establish the Relationship Using Newton's Second Law:
According to Newton's second law of motion, force is equal to mass times acceleration ($F\; =\; m\; *\; a$). Therefore, the acceleration ($a$) of a body is given by:

$a\; =\frac{F}{m}$

Thus, the acceleration of the first body is:

$a$₁ = Fm1

And the acceleration of the second body is:

$a$₂ = Fm2

3. Use the First Equation of Motion:
The first equation of motion relates final velocity, initial velocity, acceleration, and time:

$v\; =\; u\; +\; a\; *\; t$

Since both bodies start from rest ($u\; =\; 0$), the equation simplifies to:

$v\; =\; a\; *\; t$

Rearranging this to solve for time ($t$) gives:

$t\; =\frac{v}{a}$

4. Substitute Acceleration into the Time Formula:
Substituting the acceleration $a\; =\frac{F}{m}$ into the expression for time, we get:

$t\; =\frac{v}{\left(\frac{F}{m}\right)}=\frac{v*m}{F}$

Since the final velocity ($v$) and the force ($F$) are the same for both bodies, the quantity $\frac{v}{F}$ is a constant. Therefore, the time required is directly proportional to the mass of the body:

$t\; \propto \; m$

5. Calculate the Ratio of Times:
Since $t\; \propto \; m$, the ratio of the times $t$₁ : t₂ is equal to the ratio of their masses $m$₁ : m₂:

$\frac{t_1}{t_2}=\frac{m_1}{m_2}$

Substituting the given masses ($m$₁ = 2 kg and $m$₂ = 5 kg) into the equation yields:

$\frac{t_1}{t_2}=\frac{2}{5}$

Thus, the ratio of the times required is 2:5.