(d) None133. Two bodies 'A' and 'B' have velocities in theratio of 2 :

1.	(d) None133. Two bodies 'A' and 'B' have velocities in theratio of 2 : 1 mass of body A is half of the mass ofbody 'B' the kinetic energies of 'A' and 'B' are inthe ratio of
Answer» Given : • Two BODIES 'A' and 'B' have velocities in the ratio of 2 : 1. • Mass of body A is half of the mass of body 'B'. To calculate : • The ratio of kinetic energies of A and B. Calculation : As velocities of 'A' and 'B' are in the ratio of 2 : 1. So, Let , • Velocity of body A = 2v • Velocity of body B = 1v or v Also, let us ASSUME , • Mass of body B = m So, as per the given question mass of body A is half of the mass of body 'B'. $\longrightarrow$ Mass of Body A = $\sf { \dfrac{1}{2} m }$ Now, let's calculate the ratio of the kinetic energies of 'A' and 'B' . $\longrightarrow \sf { Ratio = \dfrac{Kinetic \: Energy_{(Body \: A)} }{Kinetic \: Energy_{(Body \: B)}} }$ Kinetic Energy of Body A : We KNOW that, $\bigstar \: \boxed{\sf {K.E = \dfrac{1}{2}m{v}^{2}}} \\$ • K.E = Kinetic Energy • m = mass • v = velocity Here, ✰ Mass of A = $\sf { \dfrac{1}{2}m}$ ✰ Velocity of A = 2v Substituting VALUES, $\longrightarrow \sf {K.E_{(Body \: A)} = \dfrac{1}{2} \times \dfrac{1}{2}m \times {(2v)}^{2} }$ $\longrightarrow \sf {K.E_{(Body \: A)} = \dfrac{1}{\cancel{4}}m \times {\cancel{4}v}^{2} }$ $\longrightarrow \sf {K.E_{(Body \: A)} = 1m \times {v}^{2} }$ $\longrightarrow \boxed{\sf {K.E_{(Body \: A)} = m{v}^{2}} }$ Kinetic Energy of Body B : We know that, $\bigstar \: \boxed{\sf {K.E = \dfrac{1}{2}m{v}^{2}}} \\$ • K.E = Kinetic Energy • m = mass • v = velocity Here, ✰ Mass of B = m ✰ Velocity of B = v Substituting values, $\longrightarrow \sf {K.E_{(Body \: B)} = \dfrac{1}{2} \times m \times {(v)}^{2} }$ $\longrightarrow \sf {K.E_{(Body \: B)} = \dfrac{1}{2} \times m \times {v}^{2} }$ $\longrightarrow \boxed{ \sf {K.E_{(Body \: B)} = \dfrac{1}{2}m{v}^{2}} }$ Calculating Ratio : $\longrightarrow \sf { Ratio = \dfrac{Kinetic \: Energy_{(Body \: A)} }{Kinetic \: Energy_{(Body \: B)}} }$ $\longrightarrow \sf { Ratio = \dfrac{ \cancel{m{v}^{2}} }{ \cfrac{1}{2} \cancel{m {v}^{2}}} }$ $\longrightarrow \sf { Ratio = \dfrac{1 }{ \cfrac{1}{2}} }$ $\longrightarrow \sf { Ratio = 1 \times \dfrac{2}{1}}$ $\longrightarrow \sf { Ratio = \dfrac{2}{1}}$ $\longrightarrow \boxed {\pmb{ \rm \red { Ratio = 2:1}}}$ Therefore, the kinetic energies of 'A' and 'B' are in the ratio of 2 : 1.