Sand kept inside the trolley drains out from its floor at a constant rate of 0.2kg/s. A force 15N is acting on the trolley as shown in figure. After time 5sec, which of the following option(s) is/are true?
A
Mass of trolley is 1kg
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B
Net force on trolley is 15N
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C
Velocity of trolley is 75ln2m/s
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D
Velocity of trolley is 75ln3m/s
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Solution
The correct option is C Velocity of trolley is 75ln2m/s Initial mass of trolley, m0=2kg
Rate of mass reduction, μ=dmdt=0.2kg/s
Final mass of trolley after 5s is m=m0−μt ⇒m=2−(0.2×5) ⇒m=1kg
Let, at t=5sec, velocity of trolley be v.
Since sand is drained out from the trolley, its exit velocity will be the same as the velocity of trolley.
Relative velocity with which sand leaves the trolley vst=vs−vt=v−v=0
When mass is reducing with time, then Ft acts opposite to the direction of vst
As velocity of sand w.r.t trolley is zero, Ft=vstdmdt=0
Therefore, no thrust force will act,
i.e Fnet=F ⇒ma=F ⇒(m0−μt)dvdt=F ⇒v∫0dv=t∫0Fdtm0−μt ⇒v=F[ln(m0−μt)−μ]t0 ⇒v=−Fμ[ln(m0−μt)−ln(m0)] ⇒v=−Fμlnm0−μtm0=Fμlnm0m0−μt
Substituting values of F=15N,m0=2kg,μ=0.2kg/s,t=5s ⇒v=150.2ln22−(0.2×5) ∴v=75ln2m/s
Hence, net force on trolley is, Fnet=F=15N