Question

Two bodies of masses ${m_1}$ and ${m_2}$ are acted upon by a constant force $F$ for a time $t$. They start from rest and acquire kinetic energies ${E_1}$ and ${E_2}$ respectively. Then $\dfrac{{{E_1}}}{{{E_2}}}$ is:
A. $\dfrac{{{m_1}}}{{{m_2}}}$
B. $\dfrac{{{m_2}}}{{{m_1}}}$
C. $1$
D. $\dfrac{{\sqrt {{m_1}{m_2}} }}{{{m_1} + {m_2}}}$

Answer 1

Hint: Kinetic energy is generally possessed due to the motion of the object; it is the amount of work needed to be done no that body to accelerate it for achieving the stable velocity from rest.To solve this we will make use of the formula of kinetic energy.

Complete step by step solution:
Given that,
The mass of the bodies are ${m_1}$ and ${m_2}$ on which force $F$ is applied for timet.
The acceleration generated due to that force on body ${m_1} = \dfrac{F}{{{m_1}}}$
The acceleration generated due to that force on body ${m_2} = \dfrac{F}{{{m_2}}}$
As we know, velocity will be equal to product of acceleration and time;
Velocity = Acceleration $\times$ time
Let, the velocities acquired by bodies ${m_1}$ and ${m_2}$
For mass ${m_1}$ velocity is ${v_1} = \dfrac{F}{{{m_1}}} \times t$
For mass ${m_2}$ velocity is ${v_2} = \dfrac{F}{{{m_2}}} \times t$
As we know that, Kinetic energy $E = \dfrac{1}{2} \times m \times {v^2}$
Let, the kinetic energies acquired by bodies ${m_1}$and ${m_2}$be ${E_1}$ and${E_2}$;
So, the ratio between ${E_1}$ and ${E_2}$ is
$\dfrac{{{E_1}}}{{{E_2}}} = \dfrac{{\dfrac{1}{2} \times {m_1} \times {v_1}^2}}{{\dfrac{1}{2} \times {m_2} \times {v_2}^2}}$
$\dfrac{{{E_1}}}{{{E_2}}} = \dfrac{{\dfrac{1}{2} \times {m_1} \times {{\left( {\dfrac{F}{{{m_1}}} \times t} \right)}^2}}}{{\dfrac{1}{2} \times {m_2} \times {{\left( {\dfrac{F}{{{m_2}}} \times t} \right)}^2}}}$
On solving we get
$\dfrac{{{E_1}}}{{{E_2}}} = \dfrac{{{m_2}}}{{{m_1}}}$
Hence, the correct option is B.

Note: The kinetic energy is a form of energy which arises within the body by the application of some force or work on it. It is proportional to the velocity of the moving body. It also depends on the mass of the moving objects.