Question

Two bodies A and B of masses \( 2m \) and \( m \) are projected vertically upwards from the ground with velocities \( u \) and \( 2u \) respectively. The ratio of the kinetic energy of body A and the potential energy of body B at a height equal to half of the maximum height reached by body A is:

• A. 8 : 1
• B. 1 : 1
• C. 4 : 1
• D. 2 : 1

Hint:

When dealing with energy conservation problems, always consider the total energy at different points, and apply conservation of mechanical energy to find the velocities and energies at other points.

Answer 1

The Correct Option is D

Let the velocity of body A be \( u \) and that of body B be \( 2u \). The masses of body A and body B are \( 2m \) and \( m \), respectively. We are asked to find the ratio of the kinetic energy of body A and the potential energy of body B at a height equal to half of the maximum height reached by body A. ### Step 1: Maximum height reached by body A The maximum height \( H_A \) reached by body A is given by the formula: \[ H_A = \frac{u^2}{2g}, \] where \( g \) is the acceleration due to gravity. ### Step 2: Kinetic energy of body A at half of the maximum height At half of the maximum height \( \frac{H_A}{2} \), the velocity of body A can be found using the energy conservation principle. The total energy at launch is: \[ \frac{1}{2} (2m) u^2 = 2m g H_A. \] At height \( \frac{H_A}{2} \), the potential energy is \( 2m g \times \frac{H_A}{2} = m g H_A \), and the remaining energy is the kinetic energy. Therefore, the velocity at half of the maximum height is: \[ K_A = \frac{1}{2} (2m) v^2, \] where \( v \) is the velocity at height \( \frac{H_A}{2} \). ### Step 3: Potential energy of body B at height \( \frac{H_A}{2} \) The potential energy of body B at height \( \frac{H_A}{2} \) is: \[ P_B = m g \times \frac{H_A}{2}. \] ### Step 4: Ratio of kinetic energy of body A to potential energy of body B The kinetic energy of body A and the potential energy of body B at height \( \frac{H_A}{2} \) are proportional to each other, and we find that their ratio is: \[ \frac{K_A}{P_B} = 2 : 1. \] Thus, the correct answer is option (4).