In this question, we are given that two bodies A and B have masses m and 4m respectively, and they are moving with equal linear momentum.

The linear momentum of an object is given by the product of its mass and velocity. Since both A and B have equal linear momentum, we can write:

m_A * v_A = 4m_B * v_B

Dividing both sides of the equation by m_A * v_A, we get:

v_B / v_A = 1/4

The ratio of velocities between B and A is 1/4.

The kinetic energy of an object is given by the equation KE = (1/2) * m * v^2, where m is the mass and v is the velocity. The ratio of kinetic energies between B and A can be found by taking the ratio of their masses and the square of the ratio of their velocities:

KE_B / KE_A = (4m / m) * (v_B / v_A)^2 = 4 * (1/4)^2 = 1/4

Therefore, the correct answer is option 2: the ratio of kinetic energies between A and B is 4:1.