In two systems of units- the relation between velocity- acceleration and force is given by v2-v1- 2t -a2-a1- t-F2-F1- t where - is a constant- Then- in this new system

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Question

In two systems of units, the relation between velocity, acceleration and force is given by $v_{2} = \frac{v_{1} ε^{2}}{t}, a_{2} = a_{1} ε t, F_{2} = \frac{F_{1}}{ε t}$ where $ε$ is a constant. Then, in this new system

m_{2} = \frac{m_{1}}{ε^{2} τ^{2}}

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m_{2} = ε^{2} τ^{2} m_{1}

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L_{2} = \frac{L_{1} ε^{3}}{τ^{3}}

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L_{2} = \frac{L_{1} τ^{3}}{ε^{3}}

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Solution

The correct option is A $m_{2} = \frac{m_{1}}{ε^{2} τ^{2}}$
The relation between the velocities, acceleration and the forces of the two systems is given as:
$v_{2} = \frac{v_{1} ϵ^{2}}{t}, a_{2} = a_{1} ϵ t, F_{2} = \frac{F_{1}}{ϵ t}$

We know that the force acting on the system is given as:
$F_{2} = m_{2} a_{2}$

Rearrange the above equation for the mass:
$m_{2} = \frac{F_{2}}{a_{2}}$

Substitute the values of force and acceleration in the equation:
$\frac{(\frac{F_{1}}{ϵ t})}{a_{1} ϵ t} = m_{2}$

$\Rightarrow m_{2} = (\frac{f_{1}}{a_{1}}) \frac{1}{ϵ^{2} t^{2}} = \frac{m_{1}}{ϵ^{2} t^{2}}$

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In two systems of units, the relation between velocity, acceleration and force is given by v2=v1ε2t,a2=a1εt,F2=F1εt where ε is a constant. Then, in this new system

In two systems of units, the relation between velocity, acceleration and force is given by $v_{2} = \frac{v_{1} ε^{2}}{t}, a_{2} = a_{1} ε t, F_{2} = \frac{F_{1}}{ε t}$ where $ε$ is a constant. Then, in this new system