QJEEM12Ph06ef: JEE Main 2012 Physics - Class 11 JEEM12Ph06ef Solution: A particle of mass is at rest at the origin at time It is…

Question

Verified Solution • 2025–26 Session moderate

A particle of mass

m

is at rest at the origin at time

t=0.

It is subjected to a force

F\left( t \right) = {F_0}{e^{ - bt}}

in the

x

direction. Its speed

v(t)

is depicted by which of the following curves?

A https://ik.imagekit.io/novatra/photos/2ff4fce2b7c7a82f8902a71648498fef8c1cea36114f48c330c93a636b45cd28.webp
B https://ik.imagekit.io/novatra/photos/91a7b2c3268a40ee966cfb32b235acddbed8194fa0aa198c483af3cc6d4cc1ce.webp
C https://ik.imagekit.io/novatra/photos/0f90824efac510a5ec5e27502d214a4600f5a2209410875ce823072f5a9d6fb3.webp
D https://ik.imagekit.io/novatra/photos/8442c5173e0a176a77f237fe2e0ac37b79bff3840097db967833ad8444052a9e.webp

Accepted Answer

Given that

F\left( t \right) = {F_0}{e^{ - bt}}

\Rightarrow m{{dv} \over {dt}} = {F_0}{e^{ - bt}}

{{dv} \over {dt}} = {{{F_0}} \over m}{e^{ - bt}}

\Rightarrow \int\limits_0^v {dv} = {{{F_0}} \over m}\int\limits_0^t {{e^{ - bt}}} \,dt

v = {{{F_0}} \over m}\left[ {{{{e^{ - bt}}} \over { - b}}} \right]_0^t = {{{F_0}} \over {mb}}\left[ { - \left( {{e^{ - bt}} - {e^{ - 0}}} \right)} \right]

\Rightarrow v = {{{F_0}} \over {mb}}\left[ {1 - {e^{ - bt}}} \right]

\Rightarrow v_{max} = {{{F_0}} \over {mb}}

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