39. Three different objects of masses m1, m2 and m3 are allowed to fall from rest and from the same point O along three different frictionless paths. The speeds of the three objects on reaching the ground will be in the ratio of

39. Three different objects of masses \(m_1\), \(m_2\) and \(m_3\) are allowed to fall from rest and from the same point O along three different frictionless paths. The speeds of the three objects on reaching the ground will be in the ratio of

(a)  \( m_1 \,:\, m_2 \,:\, m_3 \)

(b)  \( m_1 \,:\, 2m_2 \,:\, 3m_3 \)

(c)  \( 1 \,:\, 1 \,:\, 1 \)

(d) \( \frac{1}{m_1} \,\):\(\, \frac{1}{m_2} \,\):\(\, \frac{1}{m_3} \)

Answer :  (c)  \( 1 \,:\, 1 \,:\, 1 \)

\( \text{P.E.} = \text{K.E.} \)

\( mgh = \)\(\frac{1}{2}\)\(mv^2 \)

\(\rightarrow v = \sqrt{2gh} \)

\( \therefore \; v \text{ is same for all masses, even if } t \text{ is different.} \)

\(v_{m_1}\)\( \,:\, \)\(v_{m_2}\)\( \,:\, \)\(v_{m_3}\)\( = 1 \,:\, 1 \,:\, 1 \)