First slide

Conservation of mechanical energy

Question

A sphere of mass m, moving with velocity V, enters a hanging bag of sand and stops. If the mass of the bag is M and it is raised by height h, then the velocity of the sphere was

Moderate

A

$\frac{M + m}{m} \sqrt{2 gh}$
B

$\frac{M}{m} \sqrt{2 gh}$
C

$\frac{m}{M + m} \sqrt{2 gh}$
D

$\frac{m}{M} \sqrt{2 gh}$

Solution

By the conservation of linear momentum
Initial momentum of sphere = Final momentum of system
$mV = (m + M) v_{sys.}$ …(i)
If the system rises up to height h then by the conservation of energy
$\frac{1}{2} (m + M) v_{sys.}^{2} = (m + M) gh$ …(ii)
$\Rightarrow v_{sys.} = \sqrt{2 gh}$
Substituting this value in equation (i)
$V = (\frac{m + M}{m}) \sqrt{2 gh}$

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